Integral Calculus

Definition

Integrals are the values of the function found by the process of integration. The process of getting f(x) from f'(x) is called integration. Integrals assign numbers to functions in a way that describe displacement and motion problems, area and volume problems, and so on that arise by combining all the small data. Given the derivative f’ of the function f, we can determine the function f. Here, the function f is called antiderivative or integral of f’. [3]

Example: Given: f(x) = x².

Derivative of f(x) = f'(x) = 2x = g(x)

if g(x) = 2x, then the anti-derivative of g(x) = ∫ g(x) = x²

Integration is the opposite of differentiation! Integration is a way of adding slices to find the whole, and differentiation is a process of finding a function that outputs the rate of change of one variable with respect to another variable.

Calculus Fact

Definite integrals are about finding the area between a function and the x-axis.

Who

Anyone, like me, that can perform integral calculus (pure mathematics) but always wondered what is could be used for. In class, we talked area under the curve, maxima and minima, limits and many other topics, but I never fully appreciated the power of integral calculus and what the area under the curve meant until a number of years after I graduated college.

See Differential Calculus | Who.

What

What is an integral?

Integrals form a major pillar of calculus, and have many practical applications such as computing the area bounded by a curve and finding the distance traveled by an object directly from its velocity v(t). (See 9. Applications of Integration for more uses of integration.)

But what exactly are integrals? At a basic level, you can think of integrals as “undoing” derivatives much in the same way that logarithms “undo” exponentials. However, this is just the beginning of what integrals can do.

For example, with integrals, you can show that this shape has finite volume and infinite surface area.

There is quite a lot to unpack here, but do not fret. In Integrals in a Nutshell, we’ll get started understanding what integrals are and how they are used.

Where we start:

The graph below shows the region bounded by the graph of y = f(x), the x-axis, and the vertical lines
x = 0 and x = b.

What happens to the area of this region, A, as you change b?

It remains constant no matter what b is.
It changes steadily (i.e. at a constant rate) as b increases.
It changes, but the rate isn’t steady (i.e. the rate depends on b).

Where we end up:

We can approximate the shape under a curve with some rectangles. Adding up the rectangular areas gives us an estimate for the true area.

n is the number of rectangles, and the plot displays their total area.

Use this to estimate the blue area, and therefore the integral of f between 0 and 1.

Courtesy of Brilliant Math Weekly email What is an integral?

What is the area under the curve?

Area under the curve basically signifies the magnitude of the quantity that is obtained by the product of the quantities signified by the x and the y axes.

For example, consider a velocity-time graph and let y-axis denote the velocity of an object (in metre/second), and let x-axis denote the time taken by the object (in seconds). The area under an arbitrary curve would signify the displacement of the object (in metres). Since we know that displacement is the product of velocity and time. [1]

The area under a curve will indicate a number directly related to the data. Depending on the problem you are solving, it will be a solution to a question.

For instance, a farmer has two tractors. One uses x amount of fuel to till an acre of land, while the other uses y amount. One tractor has 1200 hp, and the other has 1900 hp, so they move at different speeds, and one can pull a wider plow than the other.

The field is not flat but has a hill in the middle. One tractor can operate on a slope up to 14 degrees, while the other can operate up to 17 degrees without tipping over. The field is not square, and it isn’t quite round.

Which tractor should he use to till which parts of the field to minimize his fuel usage?
Which tractor should he use to get the entire field planted the fastest?
How much of the field can he till and how much will remain unused and inaccessible?

I haven’t provided enough detail to solve this problem, but the practical application of calculus for everyday usage is the point I wish to make here.

One of the solutions (or the result to an intermediate step in your solution) may be derived from the area under the curve of a formula you may derive to solve the problem. [2]

In addition to the answers given, it should be always kept in mind that math is a tool to explain natural phenomenon and processes.

Integrals have two pictures to understand from

Geometric
Physical

Area under curve is a geometric interpretation, while integral of velocity function over a time interval is a physical interpretation, which is distance or displacement.

It’s easy to have a geometric interpretation up to double integrals, but its difficult (at least for me!!) to interpret higher integrals geometrically.

It’s better to always interpret maths physically (whenever possible). This makes studying math fun and interesting. [4]

The area under the curve on a distance vs time graph is known as the absement, which is a measure of “how far away” an object is AND “for how long.” In SI units, absement is measured in meter-seconds. At first glance, this doesn’t seem like it would be very useful, but absement does have many applications, including in kinesiology and modeling fluid flow. This website provides a useful example, in my opinion: The time-integral of displacement. Examples. [5]

Why

See Theoretical Knowledge Vs Practical Application.

How

Determining which integration technique works is like figuring out which puzzle piece fits. Students learn numerous integration techniques, such as u-substitution, integration by parts, partial fractions, and trigonometric substitutions. It may be easy to solve homework questions where the methods are isolated, but when all the integration techniques are mixed, identifying the correct method becomes challenging. Below are the steps or sequence of questions to ask to determine which integration method to use. [merylsmagicmath]

Is it a “basic” antiderivative? Use a standard table integral.
Can you rewrite the integrand (multiply, divide, rewrite exponents, trigonometric identities, long division, completing the square) to make it a basic antiderivative?
Is there a composition of functions? Use u-substitution.
Is there a product? Use integration by parts or tabular integration.
Is there a fraction whose denominator can be factored? Use partial fractions (a.k.a. decomposition).

(See also the videos on how to determine which integration technique to use.)

Why Use Radians in Calculus?

Mathematics often becomes simpler when we choose the right coordinate system.

In algebra, the right substitution can turn a messy expression into a simple one.

In geometry, Cartesian coordinates are good for lines, while polar coordinates are good for circles.

In probability, logarithms turn products into sums.

In calculus, radians are the right coordinate for angles.

They are not merely a convention. They are a coordinate choice that aligns angle with arc length, motion, differentiation, integration, and infinite series.

Using radians is like choosing a map whose grid follows the landscape.

Degrees are useful landmarks.

Radians are the terrain.

Leung, Barry. “Why Use Radians in Calculus?” Math Games, June 21, 2026. https://medium.com/math-games/why-use-radians-in-calculus-104c1b401344.

Many of the References and Additional Reading, websites and Videos will assist you with how to perform integration. Also, I hope I have helped you to appreciate the application of integration.

As some professors say: “It is intuitively obvious to even the most casual observer.”

References

[1] Maheshwary, Yashit. “Integration (Mathematics): What Does An Area Under The Curve Really Mean?”. 2021. Quora. https://www.quora.com/Integration-mathematics-What-does-an-area-under-the-curve-really-mean.

[2] Schmaltz, Wade. “Why Is It Important To Know The Area Of A Curve In Integral Calculus?”. 2021. Quora. https://www.quora.com/Why-is-it-important-to-know-the-area-of-a-curve-in-integral-calculus.

[3] ⭐ “Integral Calculus – Formulas, Methods, Examples | Integrals”. 2021. CUEMATH. https://www.cuemath.com/calculus/integral/.

[4] Bodkhe, Sagar. “Integration (Mathematics): What Does An Area Under The Curve Really Mean?” 2021. Quora. https://www.quora.com/Integration-mathematics-What-does-an-area-under-the-curve-really-mean.

[5] Vital Sine. “What Does The Area Under The Curve Represent On A Distance Vs. Time Graph? – Quora”. 2021. Quora. https://www.quora.com/What-does-the-area-under-the-curve-represent-on-a-distance-vs-time-graph.

[6] “5 Steps to Determine Which Integration Technique to Use.” merylsmagicmath, February 7, 2023. https://www.merylsmagicmath.com/post/whichintegrationtechnique.

This post will help you decide which method is appropriate for various integrals and how to check that your antiderivative is correct. The post explains each method in more detail, providing examples, and common misconceptions at the end.

See Also
Dawkins, Paul. “Calculus II.” Integration Strategy, November 16, 2022. https://tutorial.math.lamar.edu/classes/calcii/integrationstrategy.aspx.

Additional Reading

“Calculus Cheat Sheet”. 2022. tutorial.math.lamar.edu. https://tutorial.math.lamar.edu/pdf/Calculus_Cheat_Sheet_All.pdf.

“Calculus II Key Terms, Definitions | Vocabulary.” fiveable, Accessed July 15, 2026. https://fiveable.me/calc-ii/key-terms.

133 essential vocabulary terms and definitions to know.

DVD, Math. 2021. “Table Of Integrals In Calculus”. mathtutordvd.com. https://www.mathtutordvd.com/public/Table-of-Integrals-in-Calculus.cfm.

Guichard, David. “9. Applications of Integration.” Accessed July 14, 2026. https://www.whitman.edu/mathematics/calculus_online/chapter09.html.

“Integration Rules”. 2022. mathsisfun.com. https://www.mathsisfun.com/calculus/integration-rules.html.

⭐ “Introduction To Integration”. 2021. mathsisfun.com. https://www.mathsisfun.com/calculus/integration-introduction.html.

“Intuition Behind The Concept Of The Integral”. 2021. Medium. https://medium.com/math-simplified/intuition-behind-the-concept-of-the-integral-7f9c5e43825d.

Lloyd, Philip. “If the definite integral of a function equals zero, then what does that mean?” 2023. Quora. https://qr.ae/prgAoU.

“Mathwords: Area Under A Curve”. 2021. mathwords.com. https://www.mathwords.com/a/area_under_a_curve.htm.

⭐ Ryan, Mark. Calculus for Dummies. Hoboken, NJ: Wiley Publishing, Inc., 2003.

⭐ Strogatz, Steven. Infinite powers: how calculus reveals the secrets of the universe. Boston: Houghton Mifflin Harcourt: 2019.

⭐ Udall, Kai. “What Is The Significance Of The Area Under The Curve?”. 2021. Quora. https://www.quora.com/What-is-the-significance-of-the-area-under-the-curve.

Suppose you have a function that graphs velocity on the y axis and time on the x axis. Velocity is defined as distance over time. When we calculate the area under the curve of our function over an interval. In this case our interval would be two points of time, and our area would be the distance travelled. In simple examples this is easy to do without thinking about the area under the curve or the integral.

Learning

Sanderson, Grant. “Calculus.” 3Blue1Brown. Accessed July 22, 2026. https://www.3blue1brown.com/?topic=calculus.

Visual introductions to the core ideas of derivatives, integrals, limits and more.

Jerison, David. “Resources | 18.01 | Fall 2006 | Undergraduate.” MIT OpenCourseWare. Accessed July 22, 2026. https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/download/.

This introductory calculus course covers differentiation and integration of functions of one variable, with applications. The basic objective of Calculus is to relate small-scale (differential) quantities to large-scale (integrated) quantities. This is accomplished by means of the Fundamental Theorem of Calculus. Students should demonstrate an understanding of the integral as a cumulative sum, of the derivative as a rate of change, and of the inverse relationship between integration and differentiation.

Slap, Infinity. “Single Variable Calculus HUB.” Medium, July 18, 2026. https://medium.com/@navaneeth.penumarthi/single-variable-calculus-hub-019244708e76.

After completing the Linear Algebra series successfully, I moved on to watch the lecture series Introduction to Single Variable Calculus (18.01) by Professor David Jerison on MIT OpenCourseWare. This blog series is my attempt to write down everything I genuinely understand from those lectures.

Constant of Integration

Why is the constant of integration a constant? Since all real numbers have values that can change, isn’t it difficult to see it as a constant?

I hope this explanation is clear and easy to follow. Just observe what happens when I differentiate these functions.

Lloyd, Phillip. 2025. “Why is the constant of integration a constant? Since all real numbers have values that can change, isn’t it difficult to see it as a constant?” Quora. Accessed July 15. https://qr.ae/pAE7kz.