What is Integral? Everything you need to know about Integration

What is Integral

If you’re working in the field of mathematics or science, you might have come across the term integral. If you’re not sure what it is or if you want to know more about it, you’re in the right place. In this read, we will reveal everything you need to know about ‘What is Integral?’

Integral Definition and History

When you’re reading something, you chew it until it breaks down to smaller pieces so you can swallow the food easily. So, here we are going to break down What is Integral? into smaller pieces so you can understand it easily. Integral is a similar process.

Refer: Top Important Integration Rules to Simplify Calculus

You calculate various parameter such as area, volume, Movement, and many other parameters by breaking them down into infinitesimally small particles and then calculating the parameter for each of the infinitesimal particle. You then add the answer to all of these particles to obtain the final answer.

In Layman’s terms, calculation of values by breaking them down and adding each value is called integration or an integral.

The process of integration is not new to humans. The method of integration was first chosen by Eudoxus an Ancient Greek astronomer. He divides the original bodies to infinitesimally small particles. Since, he knew it requires parameters ( Area, Volume) of these tiny particles; he was able to calculate the necessary parameters of the large bodies more simply.

Even though Eudoxus has the first document use of the technique of integral or integration, usage of this concept by Archimedes in the 3rd century made it popular among scholars. He was successful in implementing the method of integration to find out the areas of the different parabola. Due to this, the technique of integration got a significant amount of popularity in the field of mathematics.

The technique went under the covers during the dark ages. Again in the 17th century, with the works of Cavalier, and Fermat the concept of integration was reborn as we are seeing it now. Later in the same era, Newton and Leibniz placed the foundation work for the modern calculus or the infinitesimal calculus. One can call the 17th century as the birth era of the modern calculus.

What is Integral with Limits?

Even though Newton and Leibniz gave the base for modern calculus, Riemann was the first person to formalize integration with Limits.

Applying the concept of integration in the limits made many discoveries we see today possible. However, some concepts, like Fourier Analysis, depends mostly on more general functions.

What is Integral Notation?

Integral of a function f(x) with respect to x can be as,∫f(x)dx, where f(x) is a function in terms of x.

For example, f(x) = x+2, f(x) =5x^2+9x + 23, etc. ‘dx’ denotes that the integral of the function is with respect to ‘x’.

Integral Notation
Integral Notation

If the integration is concerning a function in terms of y, you will observe ∫f(y) dy.

The above integral notation proclaims as “integral of f(x) or integration f(x)”, “integral of f(y) or integration of f(y) dy”. We use these notations so that there will be no confusion in the world between one region to the other region.

Now, on the basis of limits where the function encloses, integration is of two types: definite Integration and indefinite integration.

Definite integration corresponds to all the functions with Limits, whereas indefinite integrals do not have any limits or constraints.

Definite Integration or Definite Integral

So, are you still wondering about what is integral? As to given above, Definite integration or definite integral is integration a function in some limits or with constraints.

You often face this situation when you’re trying to find the are of a curve in a specific region. There will be no integration constant in Definite integration as there is in indefinite integration.

Notation of definite integration of a function f(x) in terms of x where the function is in the limits (a, b) is the Notation of Integral itself but with the upper limit on top of ∫ and the lower limit at the bottom of ∫.  The pronunciation of it can be as “integral over a to b f(x)” or “integration of function f(x) over the limits (a, b)”. These simple notation of integration helps in avoiding confusion among people from different places.


When you are integrating a function in the known limits, there is no need to write the integration constant ‘C’ in the final answer because the constant already covers while taking the limits. However, it is not the case of Indefinite Integral.

Indefinite Integration or Indefinite Integral

Indefinite integration or indefinite integral is known as general integration since it is the first form of integration. It is usually more general than the definite Integration which implements the use of limits in integration. You can change it into application-specific Definite integration by taking the function in some limits between -∞ to +∞. Indefinite or general integration is done in Fourier Analysis.

One can compute the definite integral using nondefinite integrals. Definite integral of ∫f(x) over the limits (p, q) = F(p) – F(q), where p and q are the limits in which the function is enclosed, and Capital f or ‘F’ is the indefinite integral of the function f(x) with respect to x.

Refer: What are Improper Integrals? [Explained Easily with Examples]

Integration classifies into three types on the basis of the number of integrals or the region of function.

They are Single Integration or line integration, Double integration or Surface Integration, and Triple Integration or Volume Integration.

• Single or line integral:

The function represents a line. Notation- ∫

• Double or Surface Integral:

The function represents a surface or a plane. i.e., the integration is done to a surface, or a surface is broken down to infinitesimally small pieces, and each of those pieces adds together. Notation- ∬

• Triple or Volume Integration:

The function represents a solid. A solid is broken into infinitesimally small pieces, and each of their volumes is being added to obtain the final result. Notation ∭
Integration under a closed surface which can compute and is represented by ∮.

So, I hope now you know what is integral? and everything else about it.


Please enter your comment!
Please enter your name here