Evaluating Definite Integrals

• According to integration definition in Mathematics to find the whole, we generally add or sum up many parts to find the whole. 

• We know that Integration is basically a reverse process of differentiation, which is defined as a process where we reduce the functions into smaller parts.

• To find the summation under a very large scale the process of integration is used.

• We can use calculators for the calculation of small addition problems which is a very easy task to do. We use integration methods to sum up many parts in problems where the limits reach infinity.

Some Elementary Standard Integrals in Integration

Different Types of Integrals in Mathematics

Till now we have learned what Integration is. There are two types of Integrations or integrals in Mathematics

• Definite Integral

• Indefinite Integral

What is Definite Integral?

• A Definite Integral has start and end values.

• In simpler words there is an interval [a, b].

• A definite integral is an integral that contains both the upper and the lower limits. 

• Definite Integral is also known as Riemann Integral.

• Representation of a Definite Integral -

• The variables a and b (called limits, bounds or boundaries) are put at the bottom and top of the S, like this:

In this article we are going to discuss what definite integral is, properties of definite integrals which will help you solve definite integral problems and how to evaluate definite integral examples.

Definition of Definite Integral

The Quantity

It is known as the definite integral of f(x) from limit a to b. In the above given formula, F(a) is known to be the lower limit value of the integral and F(b) is known to be the upper limit value of any integral.

There is also a little bit of terminology that we can get out of the way. The number a at the bottom of the integral sign is called the lower limit and the number b at the top of the integral sign is called the upper limit. Although variable a and variable b were given as an interval the lower limit does not always need to be smaller than the upper limit that is b here. The variables  a and b are often known as the interval of integration. Let’s understand the concept in a better way by solving definite integral problems.

Properties of Definite Integrals

1. Area Above - Area Below 

The integral adds the area above the axis but the integral subtracts the area below, to obtain a net value.

2. Adding of functions

The integral of  the functions f and g (f+g) generally equals the integral of function f plus the integral of the function g:

3. Reversing the Interval

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When we reverse the direction of the interval it gives the negative of the original direction.

4. Interval of Zero Length

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When the interval of the integral starts and ends at the same place, in simpler words if the limit is same then the result is zero:

5. Adding Intervals

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We can also add two adjacent intervals together, here’s the formula:

These properties will help you solve definite integral problems and how to evaluate definite integral examples.

Let's evaluate definite integral examples and solve definite integral problems.

Questions to be Solved

Question 1) Solve the following definite integral.

\[\int_{-2}^{3} x^{3} dx\]

Solution)\[\int_{-2}^{3} x^{3} dx\]

\[\int_{-2}^{3} x^{3} dx = [\frac{x^{4}}{4}]_{-2}^{3}\]

= \[\frac{81}{4} - \frac{16}{4}\]

= \[\frac{65}{4}\]

= 16.25

Question 2) Evaluate the integral given below.

\[\int_{0}^{\frac{\pi}{2}} cosx dx\]

Solution) Given, \[\int_{0}^{\frac{\pi}{2}} cosx dx\]

= \[\int_{0}^{\frac{\pi}{2}} cosx dx\]

On evaluating the given question,

= \[sin(\frac{\pi}{2}) - sin(0)\]

We know that the value of sin 0 is equal to zero and the value of sin (\[\frac{\pi}{2}\]) is equal to 1.

Therefore , putting the values ,

= 1- 0

= 1