# What is an integral?

##### 1 Answer

In mathematics, we talk about two types of integrals. **Definite integrals** and **indefinite integrals**.

Generally, an integral assigns numbers to functions in a way that can describe displacement, area, volume and even probability.

*Definite Integrals*

This type of integral relates to numerical values. It is used in pure mathematics, applied mathematics, statistics, science and many more. However, the very basic concept of a definite integral describes **areas**.

The definite integral of a function

The symbol used to represent this area

#diamond f " is called the integrand"#

#diamond a and b " are the lower and upper bounds"#

#diamond x " is a dummy variable"#

You might be wondering what

When we say the area defined by the function **net area**. The net area is not the same as absolute area.

If the graph of the function is above the x-axis, then it is said that the net area is positive. If it is below, the net area is negative. This might be harder to grasp at first. This is visualised below:

For example, say we are tasked with finding the net area under the curve

In our case,

To not complicate this answer, here's a video describing it in greater detail:

As such, it has been proven that

We can make a general case here; for every

At the same time, the video describes **Riemann sums**. These are used to compute integrals. Generally, the Riemann sum of a function

where

If we remember the general case formed earlier, about the integral of

*Indefinite Integrals*

These are represented as integrals with bounds. Let

You can think of indefinite integrals as generalisations of definite ones.

Instead of being defined by areas, volumes or something else, indefinite integrals correlate to derivatives. The indefinite integral of a function **antiderivative** and is often noted as

The **Fundamental Theorem of Calculus** bridges the gap between a function, its derivative and its indefinite integral. Basically, it says that

Now, say we want to find the antiderivative of the function

Using our former definition, what function do we have to differentiate to get

Except that this is not complete. Remember that, when differentiating a constant with respect to a variable, it practically dissapears, hence the true form of

Let

Since

Analogously, if we define

*Bridging the gap between definite and indefinite integrals*

Our previous antiderivative of

But we know that this is also equal to

This is where the connection between definite integrals and indefinite integrals is visible, stated formally below:

If

I hope this answer wasn't too intimidating.