Understanding Linear Functions in Calculus

Introduction Functions are used to describe mathematical things and can be difficult to define. The basic definition of a function can be said to be – a collection of ordered pairs of things, where the first members are fundamentally different in the pairs. A simple function can be as follows: [{1, 2}, {2, 4}, {3, … Continue reading "Understanding Linear Functions in Calculus"


Functions are used to describe mathematical things and can be difficult to define. The basic definition of a function can be said to be – a collection of ordered pairs of things, where the first members are fundamentally different in the pairs.

A simple function can be as follows:

[{1, 2}, {2, 4}, {3, 6}, {4, 18}, {5, 10}]

The above function has five pairs where the first members are 1, 2, 3, 4 and 5.

Functions usually have alphabetical letter as their names. So if we term this function ‘f’, which is the most common letter used for functions, then it will be properly written as:

f(1) = 2, f(2) = 4, f(3) = 6, f(4) = 8, f(5) = 10

Here are two definitions to keep in mind:

The entire set of first numbers in the function is called a domain and the first members are called arguments. In this particular example, the domain has 5 numbers and the numbers 1, 2, 3, 4 and 5 are the arguments of the function.

The whole set of second numbers in the function is called the range and the second members are called the values. Going back to the above function, the range also has 5 numbers and the numbers 2, 4, 6, 8 and 10 are the values of the function.

As mentioned before, the standard naming of a function is f. Thus we can explain this function in a sentence as follows:

The value of the function (f) at argument 1 is 2, its value at argument 2 is 4, its value at argument 3 is 6, its value at argument 4 is 8 and its value at argument 5 is 10.

Therefore a function can also be defined as a set of assigned values (the second numbers) to arguments (the first numbers)

This can be expanded further to say that the condition is that the first member of every pair is different; therefore each argument of the domain of function ‘f’ gets an exclusive value in its range.

The linear function and its importance to calculus

The linear function is the basic and essential function, on which calculus is based upon. This is a function that has a straight line running through the domain of its graphs.

Such a line can be determined by two points that lie on it. Look at the function [a, f{a}], [b, f{b}]. You can pick an “a” and “b” in the domain and determine this line defined by the two values f{a} and f{b}.

Let’s look at the formula for such a function.

It is possible to determine the linear function for the two values mentioned above by using the following formula.

f{x} = [f{a} x – b/ a – b] + [f{b} x – a/ b – a ]

Effectively this means that the first term is 0 when x is equal to b, and it becomes f{a} when x equals a. The second term is 0 when x is equal to a and it becomes f{b} when x equals b.

Another important aspect of a linear function is its slope.

This is defined as the ratio of the change of function f between x = a and x = b the change in x between the two arguments. The y-intercept is the point at which the line passes the y-axis.

The intercept of the line on the y axis is also an essential part of the linear function.

As we have seen, a linear function can be defined one that has a graph with a straight line, and can be described by its slope and y-intercept.

Special linear functions are often useful and they all have an important and unique property – they all have linear functions whose y-intercepts go through the point 0. Their graphs pass through the origin of the x and y axes. They are aptly called homogenous linear functions, and they all share the same property which is:

Their value at any permutation of two arguments is equal to the same permutations of their values at those arguments.

This can be explained by the following formula:

F{ak + bc} = af{k} + bf{c}

The above property is called the “property of linearity”.

NOTE: not all linear functions have this property of linearity. The property implies that once you know the value of a linear function and any two distinct arguments, then you can find the value at any other point or pair of arguments. This is not always true.

Practical applications of the linear functions

There are several real life applications of calculus linear functions. Remember that this is the most basic function on which other functions are based upon. The function is applied in various fields, such as meteorology, pharmaceuticals, engineering, and a lot more.

Whenever you have to create a graph in a straight line, no matter what the slope or y-intercept is, you are applying this basic principle.

NOTE: One should not confuse linear functions in calculus to linear equations in algebra. They have different properties even if sometimes their graphs can be identical. You can find a graph for a linear equation of algebra having the same slope and y-intercept as a graph for linear function of calculus, but they do not


Starting off by understanding this basic formula of calculus will make it very easy for you to move on and understand the deeper functions or integration and differentiation. Calculus should not be a behemoth to be feared but a friend to be understood. Try out some basic exercises on the linear functions in calculus and you will get a better grip on the topic.

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How To Factor Cubic Polynomials In Calculus

An important part of learning how to factor cubic polynomials in calculus and other forms of math is learning how to simplify.

Polynomials are algebraic expressions that have a higher degree than the standard x + 3 or y – 2. A polynomial refers to anything with a degree, or highest exponent, above 2, but usually means at least 4. Some of the polynomials you’ll see most often are cubic polynomials or expressions with a cube as their highest variable.

Often in higher math such as calculus, you’ll be asked to solve these expressions, However, if you’ve been working with higher math for any time, you know that the more complex that a math problem is, the more difficult its variables and solutions will be. One way, and often a mandatory step, in solving cubic polynomials is to factor the expression.

We’ll go over some ways to factor cubic polynomials, but first, let’s review what these are.

What is a Polynomial?

A polynomial is any expression that has multiple terms in it. Terms are variables or numbers such as x2, 3x3, and 5. In algebraic expressions, you don’t know what the variables represent so you can’t add them. You can only add or subtract terms that have the same degree and the same variable. For example, you could subtract 5x from 8x, but not from 8y.

Therefore, an important part of learning how to factor cubic polynomials in calculus and other forms of math is learning how to simplify.

Simplifying Cubic Polynomials

The first thing to remember is that not all polynomials can be factored. Many if not most of them, can, but there are a few that have no roots at which x is equal to zero.  However, most polynomials can be simplified into a single expression multiplied by a quadratic expression. For example, you might see (x2 – 2x +4)(x + 3).

It’s even possible that the quadratic equation can factor further, but we’ll get to that later. The first step to factoring a cubic polynomial in calculus is to use the factor theorem.

The factor theorem holds that if a polynomial p(x) is divided by ax – b and you have a remainder of 0 when it’s expressed as p(b/a), then ax – b is a factor. It’s a roundabout way of saying that if an expression divides evenly into a polynomial, then it follows that the expression is a factor.

One way to factor is to set the expression to equal 0, and then substitute various values of x until the equation is satisfied. Once you do that, you can determine that one of the factors is (x – whatever the number is. If it’s a negative, the expression would instead be x + the number. Subtracting a negative is the same as addition. Bear in mind that this is for only one factor.

The degree of an expression directly indicates how many factors it has. An expression leading with x2 has two factors.


Further Factoring

Dividing Polynomials

Factoring by Grouping

Factoring with a Constant

If your polynomial contains a constant – that is, not a variable – you can sometimes factor it using that number. If the constant has no factors other than itself and one, this makes your job a little harder, but not impossible. The defining factor as to whether this is a solution is whether setting the expression equal to 0 results in a true statement.

To start, rewrite the expression as an equation that equals 0. Then, look at your constant. Start by taking the first factor of the constant, which is always going to be 1. For example, if you have the expression x3 – 3×2 – 10x + 24 = 0, you can assume that the factors are +-1, 2, 3, 4, 6, and 12.

Check both positive and negative results, because negative numbers require opposite signs and any of them could be the solution. Place each term in the equation one by one until you get as many true statements as possible:

  • (1)3 – 3(1)2 – 10(1) + 24 =/= 0

  • (-1)3 – 3(-1)2 – 10(-1) + 24 =/= 0

  • (2)3 – 3(2)2 – 10(2) + 24 = 0

  • (-2)3 – 3(-2)2 – 10(-2) + 24 =/= 0

  • (3)3 – 3(3)2 – 10(3) + 24 =/= 0

  • (-3)3 – 3(-3)2 – 10(-3) + 24 = 0

  • (4)3 – 3(4)2 – 10(4) + 24 = 0

  • (-4)3 – 3(-4)2 – 10(-4) + 24 =/= 0

  • (6)3 – 3(6)2 – 10(6) + 24 =/= 0

  • (-6)3 – 3(-6)2 – 10(-6) + 24 =/= 0

  • (12)3 – 3(12)2 – 10(12) + 24 =/= 0

  • (-12)3 – 3(-12)2 – 10(-12) + 24 =/= 0

You might be wondering about checking negative factors for a positive constant. In a cubic polynomial, this is impossible because you have three possible factors. Only three positives can result in a positive result.

Uses of Polynomials

When learning how to factor cubic polynomials, it helps to think of real-life applications. One of the most pervasive in modern life is the way our electricity functions. Alternating current, which powers our homes, constantly fluctuates in voltage and current. The function used for this is described as a sine wave, which is expressed as a polynomial.

Any type of curved function, such as the curve of a roller coaster, is expressed as a polynomial. Most often, you’ll have computers to help graph the difficult equations, but knowing the basics can help you understand the concepts.

Final Thoughts

The methods we’ve listed are just a few ways you can solve cubic polynomials. Note that not every expression can be factored. For example, if you have a polynomial with no solutions when you attempt to solve for zero, you can conclude that it never touches the x-axis. The graph of a cubic polynomial, however, may have three possible solutions, or two places where it curves.

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What Is Entropy Information Theory In Calculus: Ultimate Guide

The specific entropy information theory in calculus we’re talking about refers to data systems that have random outcomes. Click here to find out more!

Entropy is the concept of chaos and disorder. It affects everything, including information. Learn how entropy works in calculus.

What Is The Entropy Information Theory In Calculus?


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You’ve likely heard of the concept of entropy, or at least the word. Entropy has different specific definitions depending on the context in which you’re talking about it, but the common thread among the definitions is that of uncertainty and disorder. Entropy in any given system tends to increase over time when not provided with external energy or change.

Entropy, simply put, is the measure of disorder in a system. If you look at an ice cube floating in a mug of hot tea, you can think of it as an ordered system. The tea and the ice cube are separate. Over time, however, the heat from the tea transfers into the ice, causing it to melt and mix with the tea and leaving a homogenous mixture. There is no way to restore it to its previous state.

Everything in existence is subject to entropy per the second law of thermodynamics. There are several other definitions of the word, but what we’re here to talk about is the entropy information theory in calculus.

What Is Information Entropy?

Information Entropy

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What Does A Stand For?

man looking at the equations

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Information is data about a system, an object, or anything else in existence. Information is what differentiates one thing from another at the quantum level. The entropy of information has more to do with probability than with thermodynamics: simply put, the greater the number of possible outcomes of a system, the less able you are to learn new information.

Following from this, it also means that a low-data system has a high amount of entropy. For example, a six-sided die when cast has an equal chance (unless you’re using weighted dice) of landing on any one of the six sides; the exact percentage of probability for any side landing face-up is 16.67 percent.

A 12-sided die would have a probability of 8.33 percent for landing face-up on any of the twelve sides. Its information entropy is even higher than that of the six-sided die.

As another example, if you have a regular coin with two sides, flipping it yields a 50 percent probability of landing on either heads or tails. The coin, because there is a higher probability for either outcome, has a lower level of entropy than does the die.

What Factor Influences The Level Of Entropy In A System?

The specific entropy information theory in calculus we’re talking about refers to data systems that have random outcomes, or at least some element of randomness. The level of entropy present in a data set refers to the amount of information you can expect to learn at any given time.

As such, the only factor that truly influences the level of entropy in a data set is the number of possible outcomes or the specificness of the information. For example, if you have a group of numbers from 1 to 10, and your only criteria is that the number is even, you only get one bit of information. It is measured in bits just as data in computers is.

Equation For Entropy

The equation used for entropy information theory in calculus runs as such:

H = -∑ni=1 P(xi)logbP(xi)

H is the variable used for entropy. The summation (Greek letter sigma), is taken between 1 and the number of possible outcomes of a system. For a 6-sided die, n would equal 6. The next variable, P(xi), represents the probability of a given event happening. In this example, let’s say it represents the likelihood of the die landing face-up on 3. Other factors being equal, you get 1/6.

Then, you multiply this by the logarithm base b of P(x), where b is whichever base you’re using for your purposes. Most often, you’ll use 2, 10, or Euler’s number e. For reference, e is approximately equal to 2.71828.

This, of course, represents only a single discrete variable with no other factors influencing it. If you have another factor that can influence the outcome, the formula changes. Now, it ends up being the following:

H = -∑i,j P(xi, yj)logbP(xi, yj)/p(yj)

In this case, you have to look at both x and y as variables and take their functions as dependent on one another, depending on how the problem is set up. You can also simply treat the two variables as two separate and independent events that occur. For example, f you were to flip two coins, you could treat x and y as each coin coming up heads or tails.

Alternately, you could assume that you had one coin coming up heads, which would trigger another coin flip whereas landing on tails would not. Then, you’d take y as the variable representing the second coin. Again, it varies depending on what you’re doing.

You can determine how much information is received from an event with a reduction of the formula to


You might notice that the log has no indicated base. This is because by default the base is 10. In computer science, when charting out probabilities, you might see base 2 used a lot because of binary systems. Base e is used in many scientific disciplines. Base 10, meanwhile, is used in chemistry and other sciences that aren’t quite as heavy on math.

Base 10, after all, is the basis of the decimal system and the one most people are used to working with.

The above formula represents the average amount of information you can expect to gain from an event per iteration. It assumes all factors are equal and there are no influencing conditions on the outcome. In other words, it is completely random within the available set of data.

How Is Information Measured?

Entropy information theory in calculus has several possible measurements, depending on what base is being used for the logarithm. Here are a few of the most common measurements:

  • Bits, for base 2

  • Shannons, for base e

  • Bans, for base 10

On occasion, you may have to convert one measurement to another.

Purpose Of Studying Information Theory

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One question that inevitably arises when dealing with higher mathematics is why it should be studied. After all, many skills in advanced math lack real-world applications, at least to the uninitiated. However, it should be noted that information and probability theory have several applications in computer science, such as file compression and text prediction.

For example, by using the entropy theory of information, you can help to code predictive text. The English language, for instance, has many different rules about what letters can occur in a sequence. If you see the letter ‘q’, you know that in all likelihood it’s followed the letter ‘u’. If you have two vowels, they will likely not be followed by a third, and certainly not by a fourth.

This is just one example. Another possible use in computer science is data compression as in ZIP files or image files like JPEGs. Information theory has also encompassed various scientific disciplines from statistics to physics. Even the inner workings of black holes, such as the presence of Hawking radiation, rely on information theory.

Even the act of learning higher math and abstract concepts can be beneficial, even without the knowledge. The act of learning forces the brain to process information in new ways, creating stronger connections between neurons and delaying the onset of cognitive decline. The more you learn, the more exercise your brain gets.

Understanding Information Theory

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To get the most out of and understand information theory, you need a solid grasp on a few other subjects. These are all unsurprisingly math-related. Calculus and statistics are the two main subjects you need to study before starting to work on information theory because you’ll need to know how to take integrals and derivatives from calculus.

Statistics, meanwhile, gives you a solid understanding of probability theory and the likelihood of events occurring. You’ll also be introduced to a few of the more esoteric variables like lambda, or some of the symbols like sigma for summation if you haven’t seen them before.

You’ll also need to have a solid grasp of algebra, such as knowing how to manipulate equations and variables. Although you should get plenty of practice doing this in the course of algebra, it’s a good idea to understand some other concepts such as how to take a logarithm. As you can see, logarithms play a vital part in several of the entropy equations.

Final Thoughts

Learning the entropy information theory in calculus is a good way to understand how probability works and how many of the data systems you encounter produce various amounts of information. If you have a background in thermodynamic studies, it can make it easier to understand the concept of entropy.

Entropy in information theory is slightly different than it is in other branches of science, but the basic idea is the same.

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How To Solve A Logarithmic Equation In Calculus

Logarithmic equations may have a variable, several variables, or a base you'll need to determine. You will use the Power Rule or other calculus rules.

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A logarithmic equation needs to be rewritten as an exponential equation for you to find the variable. This common calculus problem contains constants and expressions, and you’ll find logarithmic shortened to “log” in some written problems.  

Condense a problem with more than one logarithm by turning it into one equation. A logarithm is constructed in a way that allows it to change a math function into another math function to solve a problem. Multiplication turns into addition and division becomes subtraction. The function changes let you find the variable value.

You’ll work with some well-known calculus rules with logs, such as:

  • The Power Rule

  • The Product Rule

  • The Quotient Rule  

Exponents, Graphs, And Intercepts

You will work with many components to solve logarithmic equations. Here are a few of them, with some brief information about the function of each one.

An exponent is a small raised number next to a larger numeral in an equation. The small number shows how many times the larger number is multiplied by itself. The small raised number 2 shows that the larger number is multiplied by itself twice.  A number multiplied by itself twice can be referred to as 4 to the second power or four squared.

A variable is a symbol for an unknown number. Linear equations may have many values that can be used in place of the variable. Most variables, however, are solved with a single value.

The variable F represents the graph of a function. The graph of a function represents all points in f(x). You can also refer to the graph of a function as the graph of an equation.

When you look at the equation of a straight line (such as y equals mx plus b) the y-intercept is the location where the line goes through the vertical y-axis. In y equals mx plus b, b is the y-intercept, the slope is m, and the variable m is multiplied on the variable x.

An x-intercept appears at the spot where the graph crosses the x-axis. The x-intercept is also the point on the graph that shows the variable x as zero.

Overview Of Logarithmic Equations

A problem with one logarithm on each side of the equation that has the same base lets you use arguments that are the same. The expressions M and N are the arguments in the following description:

LogM equals LogN leads to M equals N

A problem with a logarithm on one side of the equation you can use an exponential equation to find the answer.

Logb M equals N leads to M equals bN

Example #1

Solve this problem:

Log(x) plus Log(x -2) equals Log3 (x plus 10)

Condense the log arguments on the left side into one logarithm with the Product Rule. You need one log expression on both sides of the equation. X will have a power of two, so you’ll need to solve a quadratic equation.

You get Log[(x) (x minus 2)] equals Log(x plus 10). Now condense (x) (x-2) = x squared minus 2x. Then you get Log(x squared minus 2x) equals Log3  (x plus 10)

Get rid of the logs and set the arguments inside the parenthesis to match each other.

X squared minus 2x equals x plus 10

Now use the factoring method to finish the quadratic equation. Move all information to one side and make the opposite side contain a zero value.

X squared minus 3x minus 10 equals zero

(x minus 5) (x plus 2) equals zero

Now turn each factor to zero and solve x. X minus 5 equals zero means that x equals 5. X plus 2 equals zero means that x equals negative 2.  X equals 5 and X equals negative 2 are the answers we may use. Now check the answers to see if they are correct.

Place the answers back in the first logarithmic equation to verify their validity.

For x equals 5, Log(5) plus Log3 (5-2) equals Log(5 plus 10)

Log(5) plus Log(3) equals Log(15)

This is the correct answer. X  equals negative 2 gives us a few negative numbers inside the parenthesis, and a log of zero and negative numbers in the equation make negative 2 the wrong answer.

Example #2

Solve the following problem: ½ log (X to the fourth power) minus log (2x minus 1) equals log (x squared) plus log (2).

Log without a written base has a base of 10. Base 10 is the common logarithm. Compress both sides of the equation into one log. You’ll see the Quotient Rule applied on the left side ( a difference of logs) and the Product Rule (the sum of logs) on the right side.

Pay attention to the ½ coefficient on the left side. You’ll need to use the Power Rule and bring the coefficient ½ up in reverse order.

Log (base), M to the k power equals K times log (base) M, then ½ log (X to the fourth power) minus log (2x minus 1) equals log (x squared) plus log (2)

Now use the ½ as an exponent on the left. Log (x to the fourth power) to the one/half power minus log (2x minus 1) equals log (x to the second power) plus log (2).

Now simplify the exponent to log (x squared) minus log (2x minus 1) equals log (x squared) plus log (2) Now condense log using the Product Rule on the right and the Quotient Rule on the left.

Log ( x squared over 2x minus one) equals log (2x to the second power)

It’s all right to show that if we have the same base in our equations (base 10), we can show that they are equal to each other.  Now drop the logs and put arguments inside their parentheses.

X squared over 2x minus 1 equals 2x squared

Use the Cross Product to solve the Rational Equation. Factor out to get the brief, final answer after moving all terms to one side of the equation. Make each factor equal to zero and then solve x.   X equal ¾ is one possible answer, x equals zero is the other.

Check your possible answers. X equals zero bring an undefined zero logarithm into the equation, which is wrong. X equals 3.4 is the only solution.

Fine Tune Your Study Schedule To Get Better Grades

Students who consistently get excellent grades in calculus or geometry do so because they study daily and show a steady interest in their classwork and homework. Establish a study routine by choosing a quiet place where you can read and practice solving problems at the same time each day (or at least a few times a week).

Bookmark math websites and videos containing videos that will help you understand the formulas and concepts that give you trouble, including any logarithmic equationAlong with your class notes and practice problems in your textbook, you’ll be equipped with everything you need to study more efficiently.

Contact other students in your class, tutors from the math department, or independent math tutors to help you if you’re unable to master a particular formula on your own. Even the best students need help from another student or a tutor from time to time.

Don’t expect calculus or any math class to be difficult or easy; work on the assignments without worrying about your grades or the outcome. Study for tests up to a week in advance, preferably with classmates or other math students. Trade tips and discuss different solutions and approaches to problems.  

No one fails calculus because they lack the skills or mental capacity to perform exercises properly. People fail because they are unwilling or unable to do the required work. Teachers and calculus experts suggest you study six or seven hours on weekends and a few hours each weeknight to get the best grades possible.

Anyone who only has five to ten hours a week to study and prepare for class or tests should delay calculus classes until they are ready to study more often. Calculus courses are fast-paced, and if you get behind it will be hard to get caught up on the lessons you missed or didn’t understand.

Work as hard as you can in the first month of class. Define your strengths and weaknesses, and enlist tutors or a study group if you need them. Don’t get behind and think you can catch up quickly without help. You may end up dropping the class if you don’t take charge of your studies.    

Don’t miss classes. Calculus isn’t like history or English, where catching up is hard, but not impossible, by yourself. If you miss one class, you should be able to catch up if you have a passable grasp of previous classes.

Anyone with a poor understanding of previous formulas and problem-solving methods will find that they are completely lost after missing a class or two. As soon as you feel confused, let your teacher, tutor, or study group know and ask for help.

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Area Under Curve In Calculus: How To Find It

The area under curve in calculus can be determined through the use of a specific formula that uses both definite and indefinite integrals. Learn it here!

How To Find Area Under Curve In Calculus

Figuring out how to find the area a under curve in a graph can be a simple process once you understand the formula and the surrounding pieces of information given in the problem. Finding the area is part of integration mathematics, and by using the appropriate formula, we can calculate not just the area, but any given quantity.

A typical graph has an x-axis and a y-axis, and when you add a curve to this structure, you’ll immediately see where the area under the curve lies. By finding the points along the curve, we can input them into the formula for finding the area under a curve, and solve for the final answer.

What Is Calculus?

Calculus is one kind of mathematics and deals with finding different properties of integrals of functions and other derivatives. To see these values, specific mathematical methods are used that are based on the sums of differences which are infinitesimal.

There are two primary types of calculus, which are integral calculus and differential calculus. Overall, calculus can be summed up as a system of calculating and reasoning different values as they undergo constant change.

These two kinds of calculus are connected to each other by a theory called the fundamental theorem of calculus. Both kinds of calculus use notions such as infinite sequences, convergence, and infinite series as it applies to limits that are well defined.

The calculus that is taught currently is credited to the 17th-century mathematicians Isaac Newton and Gottfried Leibniz. Calculus today is now widely used in other disciplines such as economics, engineering, and science.

Calculus is now part of the modern curriculum taught in most schools, and it can serve as a gateway to more complicated mathematical learning, and other studies regarding limits, functions, and additional mathematical analysis.

It’s possible that you may have also heard calculus referred to by other names such as infinitesimal calculus, or “the calculus of infinitesimals. Calculus has also been employed as a name for different kinds of mathematical notation such as Ricci calculus, lambda calculus, propositional calculus, process calculus, and calculus of variations.

Calculus dates back to ancient and even medieval times where it was used to develop different ideas surrounding calculations such as area, volume, and early ideas about limits. Calculus has been used to discover different values such as the area of a circle, and the volume of a sphere.

Definite Integrals And Finding The Area Under A Curve

What Is The Arclength Formula In Calculus

If you want to know about arc length formula then this guide is for you. In this section we will discuss how to find the arc length of a parametric curve.

Finding the distance from one point to another is critical for a variety of reasons. On a straight surface, this is incredibly easy to achieve. But what do you do with a curve? Well, that’s what mathematicians tried to figure out for many centuries. Fortunately, this is now a solved problem, and it has allowed us to advance our knowledge of calculus significantly.

From biblical times, the leading figures in science have wrestled with the issue. Some, like Archimedes, even came up with reliable solutions that could give good estimations, albeit not equal to the correct answer. Building upon his and others new theorems, later mathematicians have been able to make improvements and additions.

But it wasn’t until relatively recently, at least in terms of mathematics, that we figured out precisely how to calculate the length. Even as late as Newton, it still wasn’t completely fleshed out. Fortunately for you, we now have a simple and easy to understand formula to relate to and use to calculate distances of any length.

In this guide, we’ll look at what the formula is. We’ll also discuss why it’s important to know and how Fermat and other mathematicians throughout history derived it. By obtaining it, you’ll be able to understand the equation more fully and therefore, should be able to apply it with a higher degree of success and comprehension.

Why Do We Care About the Arclength?

If you want to do any advanced calculus, it’s critical that you can identify the length of a curve. In many cases, curves are infinitely long, and therefore your goal might be to find the distance between two values of x or y such that you can solve or implement the number into further equations.

In a practical sense, this formula could be for engineering. Imagine creating a bridge with a given height and span. You’d need to know precisely how long the metal should be before it’s going to bend into the specific curvature that you need to support the span and keep it upright.

Without this formula, it would be incredibly hard to figure this distance out. By using it, companies can save time and wasted materials, allowing them to calculate to within inches the length of piping or metal that they need. Like much of mathematics, there are often efficient uses that you may not think of if you don’t look into it.

Although most of us will never do this type of engineering, this formula was groundbreaking at the time of discovery. Without it, we would be unable to advance and do further calculations. It solved a roadblock that has allowed us to discover more advanced functions and theorems that the leading scientists are still using today.

Students often ask what the point of the formula is. You could ask the same about the Pythagorean theorem. But as you’ll see when we derive the arclength below, without Pythagoras is would be impossible for us to have discovered how to calculate the length of a given curve or arc.

Truthfully, what most people misunderstand about mathematics is that it’s a story. Teachers and students alike make the mistake of jumping forwards and backward and then complaining that it’s difficult and they can’t understand it. It’s necessary to step to step, in a logical order that the leading mathematicians took so that you can follow every part of the story.

For example, if you don’t understand graphing, you cannot understand calculus. Without algebra, you won’t grasp graphing and so on. Therefore, if you’re struggling with arcs, go back to graphing and keep going back until you get to a point where you’re comfortable. Then, work forward and don’t skip a step. Fix the foundations so that you have a better understanding.

What Is The Arclength Formula?

The arclength formula allows us to figure out the distance between two points on a curve that isn’t straight. We can also do the same for a circle or different shape. But the primary use of this equation is to solve for distances between irregular and complex curves. Here’s a quick history of the formula:

  • It’s been worked on for centuries

  • Early mathematicians like Archimedes had crude alternatives
  • Fermat and another mathematician simultaneously were working on the problem
  • Femat found fame for his discovery

  • Newton and others made improvements later

Deriving the Arclength Formula

Imagine that we have an arc on a graph, whereby the gradient is changing at a constant rate to create a smooth curve. How would we go about finding the length of the arc? This problem is what led to the derivation of the arc length formula which allows us to calculate it.

Pierre de Fermat, arguably the greatest mathematician in history, was the one who discovered and popularized this method. His work built upon that of Archimedes and other mathematicians before Fermat’s time. His idea was that by taking a tangent, you could create a formula for the line and then use small segments to estimate the length of the curve.

If we take two points, x0 and x1, on the curve, we can create a right-angled triangle to connect them. The longest side of the triangle will not perfectly follow the curve, but it’s close enough to get a reasonably accurate estimation for the length of the curve between the two points. Where S1 is the distance on the curve and x and y are points on the curve respectively.

S1 = √ (x1 − x0)2 + (y1 − y0)2

In this example, we use the mathematical symbol  Δ (delta) which means the difference between two values, in this case, x0 and x1 and y0 and y1.

S1 = √ (Δx1)^2 + (Δy1)^2

Imagine that instead of just using one large triangle, we used a lot of small ones. The result would be that we could get the longest side on each triangle closer to the correct path of the arc. Therefore, we will be able to get a much closer estimation of the true length of the curve on the graph.

S2 = √(Δx2)^2 + (Δy2)^2

S3 = √(Δx3)^2 + (Δy3)^2

Sn = √(Δxn)^2 + (Δyn)^2

Rather than having a bunch of equations, we can simplify them all into a single formula. By doing so, we can look at the curve in its entirety by segmenting it into an infinite number of infinitely small segments.

Arclength Formula

S ≈  Σ √(Δxi)^2 + (Δyi)^2


This formula above looks complicated, but it’s incredibly simple. All it’s saying is that the length S is roughly equal to the sum of all of the longest sides of the triangles, where we use n number of triangles. The problem is that it will take us years to add up all of those numbers! Hmm, there has to be a better way.

Fortunately, there is, and this is what Fermat discovered and why he’s remembered as such a talented mathematician. This calculation seems simple now because we can find it online, but this discovery was groundbreaking and change the direction of the universe for good. Without his genius, we wouldn’t be able to do much of the engineering we do today.

The solution is to make all of the Δxi in the formula be the same, so we can remove them from inside of the square root and create an integral.

To achieve this, we need to divide and multiply Δyi by Δxi, leaving us with:

S ≈  Σ √(Δxi)^2 + (Δxi)^2(Δyi/Δxi)^2


Next, we must factor out the (Δxi)^2:

S ≈  Σ √(Δxi)^2 + (1 + (Δyi/Δxi)^2)


Take (Δxi)^2 out from inside of the square root:

S ≈  Σ √(1 + (Δyi/Δxi)^2) * Δxi


We now that as n approaches infinity, we get closer to an infinite number of slices over a given distance. Therefore, the segment that each cover gets smaller, approaching an infinitely small size. As a result, we can get a more accurate estimation of the length of the arc, to the point of it becoming equal rather than approximate.

S =  lim Σ √(1 + (Δyi/Δxi)^2) *  Δxi

N→∞    i=1

Now, we have an integral. Typically we replace Δx with dx and likewise with dy. The reason for this is to symbolize that we are approaching a zero width and therefore can assume equality.

S =  lim Σ √(1 + (dy/dx)^2) *  dx

N→∞    i=1

We know that dy/dx is the derivative of the function f(x) and therefore you can substitute that into the formula if you wish. However, for most people, it’s easier to remember the above the arclength formula so that you can more easily solve the equation and use it for calculations.

In a few short steps we’ve been able to derive the formula so that rather than needing to add up a bunch of slices for an approximate answer, we can calculate a precise number. The only thing you need to do is to solve the differential and integral for the curve or arc which will allow you to find the distance and also other figures like space under the curve.

It’s important to remember that the integral will also work concerning y. This fact is useful if we already know that x = f(y) and therefore can solve.

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Mean Value Theorem for Integrals: What is It?

In this calculus guide, we are going to help you understand the mean value theorem for integrals. If you are having difficulties with it, read this!

Image from Pixabay

The mean value theorem for integrals is a crucial concept in Calculus, with many real-world applications that many of us use regularly. If you are calculating the average speed or length of something, then you might find the mean value theorem invaluable to your calculations.

Ultimately, the real value of the mean value theorem lies in its ability to prove that something happened without actually seeing it. Whether it’s a speeding vehicle or tracking the flight of a particle in space, the mean value theorem provides answers for the hard-to-track movement of objects.

What is the Mean Value Theorem?

Based on the first fundamental theorem of calculus, the mean value theorem begins with the average rate of change between two points. Between those two points, it states that there is at least one point between the endpoints whose tangent is parallel to the secant of the endpoints.

A Frenchman named Cauchy proved the modern form of the theorem. One of the most prolific mathematicians of his time, Cauchy proved the mean value theorem as well as many other related theorems, one of which bears his name.

It is also possible for a function to have more than one tangent that is parallel to the secant. The derivative, or slope, of each tangent line, is always parallel to the secant in the mean value theorem.

As an addition to the mean value theorem for integers, there is the mean value theorem for integrals. This theorem allows you to find the average value of the function on at least one point for a continuous function.

Stipulations for this theorem are that it is continuous and differentiable. That means that the line acts as a traditional function, without any odd stops, gaps, drop-offs, or any other non-continuous feature.

It also must be differentiable, which means you can find the slope of a point on the function. For cube roots or the absolute value of x, you cannot find a derivative because they are either undefined or not tangential to the average rate of change. 

Floor and ceiling functions also do not have derivatives because they are not continuous functions. Thus the mean value theorem of integers does not apply to them.

Why Is the Mean Value Theorem for Integers Important?

Like many other theorems and proofs in calculus, the mean value theorem’s value depends on its use in certain situations. Since this theorem is a regular, continuous function, then it can theoretically be of use in a variety of situations. Any instance of a moving object would technically be a constant function situation.

Real-world applications for the mean value theorem are endless, and you’ve probably encountered them either directly or indirectly at some point in your life. 

One of the classic examples is that of a couple of police officers tracking your vehicle’s movement at two different points. You are then issued a ticket based on the amount of distance you covered versus the time it took you to complete that distance. 

You were not speeding at either point at which the officer clocked your speed. But, they can still use the mean value theorem to prove you did speed at least once between the two officers. 

More specifically, consider modern-day toll roads. These roads have cameras that track your license plate, instantaneously clocking your time spent on the road and where and when you exited and entered. Law enforcement could quickly begin to crack down on speeding drivers on these roads, by merely finding the average rate of change between the two points. Drivers could then blame the mean value theorem of integers as the reason for their ticket.

Further use occurs in sports, such as racing. When investigating the speeds of various racing objects, such as horses or race cars, technicians and trainers need to know the performance of horses or race cars at specific points during the race. 

Using data obtained throughout the race, individuals can determine how their horse or car was performing at certain times. For horses, this can mean altering training patterns or other variables to improve performance related to results. Race car drivers can use the data to tune equipment in various ways to better utilize the car’s speed.

The critical part of the theorem is that it can prove specific numbers. It can determine the velocity of a speeding car without direct visual evidence, or the growth, length, and myriad other instances where an object or thing changes over time.

How to Use the Mean Value Theorem

In the real-world, a continuous function could be the rate of growth of bacteria in a culture, where the number of bacteria is a function of time. You could divide the difference in the number of bacteria by time to find out how fast they multiplied.

Applying the mean value theorem to the above situation would allow you to find the exact time where the bacteria multiplied at the same rate as the average speed. This might be useful to researchers in various ways, to determine the characteristics of certain bacteria.

Another more practical situation would be to determine the average speed of a thrown baseball. The distance of the ball thrown is a function of time. Dividing the difference in the length by the time it took for the ball to get from point a to point b would tell you how fast the ball goes.

When the mean value theorem is applied, a coach could analyze at which point the ball achieved the average speed. If the speed was faster before or after the tangential point, then the coach could alter the mechanics or delivery of the player’s throw. This would make for more optimal speed with the throw reaches the batter.

Finally, let’s find the average speed of the vehicle and then at which point during the drive, the car reached a speed equal to the average rate.

  1. 1

    First, find the total distance traveled by the vehicle. To do this, check the odometer before and after driving. Calculate the difference between the two readings.

  2. 2

    Determine the amount of time spent driving the car between those two points.

  3. 3

    Divide the distance by the time. Let’s say it’s 40 mph.

  4. 4

    Now you need to find the point – or points – during which the car was traveling at 40 mph.

  5. 5

    Graph the function.

  6. 6

    The point at which the vehicle traveled 40 mph will show as the highest or lowest point on the slope connecting the beginning of the drive and the end.

  7. 7

    Using the graph, you can then find the exact time at which the car was traveling at 40 mph.

When using the mean value theorem in practical applications like vehicle speed, it is essential to note that the average rate of change is just that – an average. If your vehicle speed is 50 mph, then at some point during your drive you drove over and under 50 mph. Of course, you would hit that speed at least twice at a minimum.

Another exciting application of the mean value theorem is its use in determining the area. When the point at which the tangent line occurs is understood, draw a line from the new point parallel to the x-axis. This line is the top of your rectangle. The bottom is the x-axis. The left side is the y-axis, and the right is the endpoint of your continuous function.

Once this is complete, the area of your rectangle will be the same as the area beneath the curve of your function. One practical application of this instance is determining the exact height of a liquid in a container. If the liquid is suspended or not at rest, then calculating the mean value theorem of integers for the endpoints of the liquid will help you to determine the resting volume.

Determining amounts of liquid or the properties of a substance are just a few of the many applications of the mean value theorem. All fields of science use this theorem, and merely finding the volume of a liquid at rest is just scratching the surface. As sport becomes more science-based, the value of this theorem will only continue to increase.


While a fundamental calculus theorem may not change your life, it can make your life a tiny bit more manageable. Understanding the movement of an object and the properties within that movement can help you make a variety of educated conclusions. 

When working in scientific fields such as physics or biology, the use of the theorem can aid in the research of particles or microscopic organisms. In sports, you can use the theorem to develop a better understanding of fast-moving objects. On the highway, the police can issue more speeding tickets. The mean value theorem of calculus is an invaluable tool for all types of people.

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How To Find The Square Root Of X Graph

How to find the square root of x graph is easier for some students than many other pre-calculus subjects because visual representation is involved.

Image source: Pixabay.com

You’ll encounter the square root of x graph in your calculus and geometry classes. Graph-related work is easy to understand even if you don’t have much-advanced math experience. Draw graphs to accompany equations to help you find answers without a struggle.  

Learn all you can about square roots before using graphs to determine solutions to assigned problems. A square root is a number multiplied by itself. Square roots are used to take measurements of spaces and buildings. There are also many other scientific and real-world uses for square roots.  

What are Square Roots?

Square roots are indicated by putting a two (commonly called squared, or multiplying the number by 2) above the number you’re multiplying.

If you calculate a number’s square root, you need to find the number that was squared in the first place. For example, three is the square root of nine. Three squared equals nine and the square root of nine equals three. All squares and square roots are displayed in pairs.

There are many different square roots, including decimal squares, perfect squares and square roots for negative numbers.

Square roots of most numbers aren’t whole and easy to determine. You probably know the square roots of four and nine by heart, but what about five or seven? The square root of seven is 2.64575131106!

You will use a calculator or table to determine most square roots. However, some teachers may want you to figure it out long-hand – or at least know the manual process for determining odd square roots.

You can figure out square roots by hand by:

  • Adding a zero to the front of numbers with an odd amount of digits before the decimal place
  • Adding a zero to the end if there is an odd number of digits behind the decimal place
  • Making sure you have an even number of digits to work with

Once you’ve adjusted the numbers, draw an area diagram and use it to determine the square root. You probably won’t need to use this method often, but it will help you understand the method behind determining square roots. Your teacher may recommend other manual methods.

Square roots are used by factory workers, engineers, and scientists. Square roots are used to calculate volume and area. When you are looking for a new house or apartment, square roots help you determine the real size of a room. A 400 square foot apartment equals 20 feet by 20 feet, for example.  

Carpenters use square roots to determine building and room measurements.


You can draw graphs on your computer or Smartphone. Go old-school and use graph paper or a calculator to solve problems if you prefer. A visual representation of the equation always helps you to calculate the answer more efficiently.

Define the variable in your equation and locate the point on your graph for the function. Buy a graphing calculator if you need to solve lots of square root graphing problems. A graphing calculator shows plotted graphs and has tangent, sine and cosine capabilities.  You can buy these calculators online or at Staples or Office Depot.

Example #1

Here’s an example of how to find the square root of x graph that involves a table. Determine the domain for this equation, determine the table for the values of the f function, and graph it. Then figure out the range.

The equation is:  f(x) equals the square root of (x squared negative 9). Find the function’s domain with x squared negative 9 is greater than or equal to zero. The domain is shown by negative 00negative 3 U (3 plus 00).

Now find values of x in the f domain and make a table of values. Consider that f(x) equals f (negative x). Check the graph symmetry and the y-axis to determine the answer.

F’s range is indicated by (0 plus 00).

Example #2

Determine the graph for the radicand (the number under the radical symbol) and devise a table of values of function f, and graph f and determine its range.

F(x) equals the square root of x squared plus 4x plus 6. You complete the square by rewriting the numbers under the square root as follows:

X squared plus 4x plus six equals (x plus 2) squared plus 2

The expression underneath the square root is positive. Therefore, f’s domain consists of all real numbers. The graph of (x plus 2) squared plus 2, or a parabola. The graph of f has the same axis of symmetry. X equals two the vertical line, is above the graph. The table of graphs can be written numerically as:

X equals the square root of (x plus two squared plus 2). On the graph, negative 2 equals 1.4, zero equals 2.4, two equals 4.2, and four equals 6.2

The f range is shown by the interval the square root of 2 plus 00.

Example #3

Here’s something a little different. Show the graph of y equals the square root of x minus one plus two as an extension of the parent graph y equals the square root of x.

Draw a graph representing y equals the square root of x. Then move the graph a unit to the right, and you’ll get y equals the square root of x minus one. Move the graph of y equals the square root of one up two units, and you’ll get y equals the square root of x minus one plus two.

The visual representation of the graph makes it easy to see the answers. Now you can figure that x is greater than or equal to one is the domain of the function y equals the square root of x minus one plus two.

The graph also shows the range of the function y equals the square root of x minus one plus two is greater than or equal to two.  

How to Study

Most people have a hard time studying for calculus, geometry or algebra exams. Most problems can be attributed to math anxiety, a common classroom ailment. Math is only hard if you think it is – like any other subject, your attitude will make or break your study habits.

Whether you are studying how to find the square root of x graph or something else, you’ll encounter obstacles if you don’t break down the process into manageable steps.

All-nighters won’t garner the best results in any school subject, and math is the hardest subject of all for last-minute study. Plan your study schedule even if you don’t have a test coming up –  daily study is the only way to familiarize yourself with calculus formulas and definitions.

Make sure you get enough sleep and eat right, especially the day before an exam. If you’re bleary-eyed and running on caffeine and sugar, you won’t be able to concentrate on the formulas and terms you need to solve problems.

You build on one theory and set of formulas in algebra, geometry or calculus before you can move on to another, more complicated aspect of advanced math. You won’t get answers on the first try unless you’re a math genius or you’re working on simple problems. Be prepared for some trial and error.

You have several options besides solo study and reviewing class notes to help you study. Need extra help? Consider the following options:

  • Math tutor

  • Study groups

  • Online math sites

  • Extra textbook problems

There are many video tutorials on YouTube and math websites to help you if written explanations are confusing. Check websites for colleges and universities, as many math departments include PDFs of handouts from some of their math classes.

Don’t ignore concepts or problems you don’t understand and move onto another lesson. You need to fully grasp one concept before moving on to the next chapter or function. Practice on as many problems as you can, and pinpoint where you made the error in your thinking.

Always ask your teacher, tutor or classmates to help you find real-life applications for math problems. When you are aware of how to use calculus or other math formulas in the real world, you’ll be less apt to have math anxiety or complain that calculus is useless and won’t help you when you graduate.

Calculus is used in manufacturing, science, engineering, astronomy, aviation, and other industries. Try to find “common ground” with your math assignments. Look for how you can use the formulas in your daily life, or a hobby or your chosen field.

Keep in mind that most math and calculus problems can be solved in several different ways. There might be a second or third way if one method is hard for you to understand. If your book or teacher only suggests one way to solve a problem, look for other methods online or ask a friend about alternate methods.

Control your math anxiety by having a set schedule for homework. Always have at least one other trustworthy person to help you review assignments and help you study for tests. Keep a positive attitude about math classes; don’t dread them.

Be open to new study methods and learn about practical ways to use calculus and other advanced math in your life outside the classroom.  

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Trapezoid Calc: Find P

Learning to solve trapezoid calc: find P problems is far easier if you learn first to do it algebraically instead of relying on a graph.

A trapezoid, also called a trapezium in some countries, is a geometric shape with one pair of  parallel sides. It looks like a complex shape when you’re trying to figure out things like area, perimeter, and angles, but it is actually quite simple when you break it down into its basic part, then use a few mathematical formulas to derive other quantities.

Perimeter, as you may recall from geometry, is nothing but the sum of the length of all the sides of a shape.

There are some instances in geometry, algebra, calculus, and practical applications that will require you to find the perimeter of a trapezoid when you may not necessarily have all the quantities. When doing a trapezoid calc: find P operation, you can add up the known sides and figure out the rest.

Properties of Trapezoids

The only constant property of trapezoids is the existence of a single pair of parallel sides. Beyond that, a trapezoid can take any configuration. The two parallel sides can have any difference in length, or they can be the same length. In other words, a rectangle or a square is technically a trapezoid, just a special type of one.

Finding the Perimeter of a Simple Trapezoid

To find the perimeter of a trapezoid (or any shape except an ellipse or a circle) you just have to add the lengths of all the sides. However, if you don’t know the side lengths, you might have a problem. By using properties of geometry, trigonometric identities, and basic logic, you can infer or find the values of the unknown sides of a trapezoid.

For example, if you know the angle formed by the base and one of the sides, and if that angle is the same on the other side, you can then infer that the opposite side must be the same length because it has to cover the same distance to reach the parallel top of the trapezoid. However, you may not be given this information outright and you’ll have to figure it out on your own.

The best way to think of a trapezoid is as a rectangle with two right triangles on the sides. Then, by using the properties of right triangles, you can figure out the length of the unknown side.

We’ll start by working with a trapezoid ABCD, where each letter represents a vertex where two line segments intersect. We’ll assume, that line segment AB and line segment CD are parallel. AB has a length of 5, and CD has a length of 11. We also know that the trapezoid has a height of 4, if you were to take a perpendicular line connecting AB to CD.

Finally, angles ACD and BCD are given to be the same.

Because we have the top base has a length of 5, you can think of the bottom base as including a rectangle with a length of 5. Because we know that the angle is the same for both sides, you know that the two sides have to be the same length. Therefore, you now have to imagine the trapezoid broken down into other shapes.

The height is 4, and the total length of the bottom base CD is 11. Because AC and BD are the same lengths, it makes things easier. Now, on each side of a length of 5 on CD, you have 6 unaccounted for, so you can think of a right triangle with 3 on one leg and 4 on the other because you already know that height.

The next step in trapezoid calc: find p is to work out the length of the right triangle’s hypotenuse. The formula for the Pythagorean theorem is a2 + b2 = c2. Square the lengths of the two legs.

The square of 3 is 9, and the square of 4 is 16. Adding these up, you get 25. Now, take the square root, which is 5. In this example, we’ve used the simplest right triangle there is.

It won’t always be this simple, but it serves as an example. Because you know that the two right triangles are the same dimensions, you can infer that the hypotenuse is the same on both sides.

Finally, you can go back to your original trapezoid and substitute the two values into the original shape. Then all you have to do is add the sides as you normally would.

In this example, 5 + 5 + 5 + 11, for a total of 26. Therefore, the trapezoid in our example has a perimeter of 26.

Using Trigonometric Identities

There may be times when you have to use some knowledge of trigonometry and a scientific calculator to help you out.

Here are the three identities you’ll use most often:

  • Sine = opposite/hypotenuse

  • Cosine = adjacent/hypotenuse

  • Tangent = opposite/adjacent

The tangent is also equal to the sine divided by the cosine.

You can also use the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C).

This is how you’ll find any unknown angles or, with the help of some algebra, the sides if you need them.

We’ll assume for now that you’re just working with a right triangle with another known angle of 60 degrees and a hypotenuse 5 units long, and you want to find the lengths of the other sides. Remember that as per geometry, the three angles of any triangle add up to 180 degrees. Because one of them is by default 90 degrees, you have 90 degrees to account for.

You know your angles: a/sin(60) = b/sin(30) = 5/sin(90). Simplifying, you get b/0.5 = 5. Therefore, b equals 10. You can get the other sides this way as well.

Finding Perimeter of a Trapezoid on a Graph

Having the figures sketched out in geometric diagrams is one thing, but having it on a graph coordinate system can make it even easier to find the perimeter. Assuming that each increment of x and y is one unit, the only part for which you might have a problem is the hypotenuse. However, even that is easy to calculate.

All you have to do is count the units of x and y that make up the right triangle, then square the units and take the square root of the result. This gives you the diagonal of the trapezoid. To get the lengths of the sides, all you have to do is count the units down to the x-axis and across to the other parallel line.

What if You Don’t Have a Graph?

Pencils, eyeglass and graph notebook

Image by marijana1 from Pixabay

Even if you don’t have a graph and are just given points, you can do simple subtraction. For example, take the points (0, 3) and (0, 9). To determine the length of the line segment and the slope, you use the formula (y1 – y0)/(x1 – x0). Substituting the values, you get 0 and 6.

The formula above is meant for slope, but it’s also an easy way to visualize. If your x-values are the same, you have a vertical line.

Now, you need to find your diagonal line of the trapezoid. It doesn’t matter what slope it is; all you’re looking for is the length. By finding the change in y and the change in x, you get two sides of your right triangle. You should have a good idea of what to do from here: square the lengths of the two sides, add them, and take the square root of the result.

Now, unless you were fortunate enough to get a rational root – i.e., one that simplifies to a fraction or decimal, you’re not going to be able to write it as a standard integer. For example, say that you got a diagonal with a length √2. You would have to add the other sides on their own, such as 3 + 3 + 5, which gives you 11. Then, you add √2.

From here, there’s nothing further you can do. If you have something like √8, you can simplify that into 2√2, which is √2 * √4. Always reduce math expressions to their simplest forms when you can. If you’re in school, odds are you’ll get only partial credit if you don’t simplify your answers.

Learning to solve trapezoid calc: find P problems is far easier if you learn first to do it algebraically instead of relying on a graph. Not all graphs will resolve to neat points and therefore risk giving you inaccurate answers.

Final Thoughts

The perimeter of a trapezoid is one step in helping you to find its area. Because the area is a function of height and width, you need to gather as many concrete measurements as you can. If you’re working on a graph, it might help to visualize the trapezoid as lying sideways, with the vertical lines being the bases and the horizontal lines being the height.

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Trapezoid Calc: Find A

Trapezoid calc: find a is easy. However, it does take some manipulation of variables and measurements when you get into more complex expressions.

A trapezoid, as you’ll recall from geometry, is a shape with two parallel sides. That’s the only definition and the only set of properties that it has. There are special trapezoids, but you might also know them as rectangles or squares. You’ll need to be able to determine the area of a trapezoid in various studies like geometry and calculus, as well as use it in practical applications.

For starts, the area of a trapezoid is helpful in calculus when getting the approximation for integrals, assuming you don’t want to go through the trouble of taking integrals directly and can just provide a rough approximation. There are several things you can do when faced with trapezoid calc: find A math problems.

What is Area?

Area is nothing but the space outlined by a two-dimensional shape, expressed in square units, such as in2 or cm2. Multiplying the length by the width provides you with this information for simple quadrilaterals, but it doesn’t work as well with trapezoids. Therefore, you have to be a little creative and be diligent about separating your trapezoid into other shapes.

Here’s what you need to find a trapezoid’s area:

  • Absolute maximum

  • Lengths of two parallel bases

  • Height

  • Dimensions of triangular sections if one of the bases is not available

Dividing Up a Trapezoid

The simplest form of an area is length times width, but this only works for quadrilaterals like squares or rectangles whose sides all make right angles. A trapezoid can be thought of as a rectangle with a triangle attached to either side, so you’ll have to find the area of each of these sections separately.

The formula for the area of a triangle is ½ bh, where b is the base and h is the height. Therefore, if you know the extra length that isn’t a part of the basic quadrilateral, as well as the height, you can easily calculate the area of the trapezoid by adding each section.

Let’s start with a simple example. Assume that you have a trapezoid ABCD with all known lengths. AB, the top line, is 5 units long, and CD the bottom line, is 8 units long.

You also know that the height is 3 units.

To start, take the basic area of the quadrilateral in the form of 5 * 4, or 20 square units. Then you’ll have to take the areas of the two triangles. Unlike with perimeter, you don’t need to know the length of the hypotenuse. All you need is that extra length and the height. Because the total is 8 and the part you’ve already calculated is 5, you have 3 units left over.

Multiply 3 by 4 to get 12, then divide by 2 to get an area of 6. Multiply by 2 to account for both sides, leaving you with 12 square units, and adding this measurement to the 20 square unit you calculated earlier leaves you with a final answer of 32 square units.

What if You’re Missing Sides?

If you’re missing a side measurement, it makes it harder to complete a trapezoid calc: find A operation, but not impossible. You’ll usually have at least one angle, and if you have an angle, you can find sides using the Law of Sines. You’ll also have at least one or two sides. It’s rare to impossible that you’ll be asked to solve a trapezoid with no measurements.

This is a simple ratio: a/sin(A) = b/sin(B) = c/sin(C)

Now because your basic part of the trapezoid is a quadrilateral, you already know that the rest of it is composed of right triangles. Let’s also assume you have a hypotenuse, side c, of 5. As such, you know that one of your angles is already accounted for. Therefore, a/sin(A) = b/sin(B) = c/sin(90). The sine of a 90-degree angle is 1.

Now, let’s assume that b, your base side, is 10 units. So, you can now put that in as well and cross- multiply: 10/sin(B) = 5. Simplifying the equation gives you 2/sin(B) = 1, or sin(B) = ½, or 30.

Plug 30 into (B) to continue with the equation to have your other value present. Also, remember from geometry that all three angles of any triangle add up to 180 degrees. Therefore, you can subtract and figure out that your missing angle is 30 degrees.

By using the Law of Sines, assuming you have a known side (it’s rare that you’ll have all sides or all angles missing), you can then take the sines of the angles you have, cross-multiply, and simplify.

Once you do that, you can take the side of the base of the triangle and the height and use those two measurements to calculate area as you normally would: ½(bh). Do this for both sides, add them, and then add the rest to the quadrilateral base. You should have this in your formula; if you don’t, it may not be possible to solve for the area of a trapezoid.


Now we can get into a slightly tricker subject: using the area of a trapezoid on a graph, such as you would do when approximating integrals. Your first step is to decide how thin you want the trapezoid to be on the graph. The more you have, the more closely you can approximate the area under the curve. For simplicity, try starting with slices with a width of 1.

This corresponds to a change of 1 for x on the graph, so this should make it easy to calculate.

Using the relevant function, such as 2x + 3, use x to find the appropriate values of y In this case, if you start at x = 1, you’ll have the vertices (0, 0), (3, 0), (0, 3), and (3, 7).

In a way, this makes it simpler to calculate because you have the points plotted out and can, therefore, find your lengths. You have an imaginary line traveling across the graph at y = 3, dividing your trapezoid into a quadrilateral and a right triangle. You should have 3 increments of x and 4 for y.

As always, you’ll find your base area first. Then find the dimensions of your triangle and find its area. Then, just add the two numbers.

How is This Useful?

If we take multiple trapezoidal segments of an integral, calculate all their areas, then add them together, we can easily approximate the area under the integral without having to go through the steps of doing derivatives and solving regarding calculus.

This sort of function has many applications in science, such as determining the total volume of liquid given a certain rate of flow over time. Of course, all this works only when the value for y is not at or below zero, at least as far as practical applications are concerned. Few, if any, practical applications use negative numbers.

Trapezoid calc: find A Another Method to Find Area

Another way that you can do trapezoid calc: find A equations is to use the formula [(b1 + b2)/2]h, where you take the average of the two bases and then multiply by the height. However, this requires you to know the bases, which isn’t always possible. If you know both of them, you can use this formula. If you only know one, you’ll use the alternate method; that is, sectioning it.


If you use the basic formula for the area of a trapezoid [(b1 + b2)/2]h, you’ll need to do a little visual tweaking. Functions travel along the x-axis of a graph, meaning that the trapezoids used to approximate them are sideways. In this case, you need to envision the graph as being sideways. The height of the trapezoid, in this case, lies parallel to the x-axis.

Meanwhile, the bases are vertical. Even though it looks vice-versa, don’t let it throw off your formula. You’re essentially treating the trapezoid as a rectangle with distorted sides.

Irrational Area Measurements

Some of the example problems we’ve covered have side measurements that work out to neat and clean integers. This isn’t always going to be the case. Because you’re dealing with the Pythagorean theorem to find the measurements of a right triangle, you’re likely going to have to deal with square roots that don’t simplify. An example is √6. You cannot simplify it any further.

Something like √12, however, can be reduced to √4 * √3, or 2√3. If you see a number like this, keep it as is unless your instructor specifically asks you to use a calculator and round to 3 or 4 decimal places. Because these square roots are irrational, you won’t get an exact match if you try to check your work. You might get something like 2.9998.

Final Thoughts

Before you start with your trapezoid calc: find A work, take a look at the problem and determine what data is present and what you need to find. This will help you determine which method you need to use to find the shape’s total area. Many times, you’ll be able to use the basic trapezoid formula. If you don’t have all the data, you’ll need to split the shape into pieces.

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