Evaluating Definite Integrals

What is Integration in Maths?

  • 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.


If \[\frac{d}{dx}\] (F(x)) = f(x), then


 ∫f(x)dx = F(x) + c


The function F(x) is called anti-derivative or integral or primitive of the given function f(x) and c is known as the constant of integration or the arbitrary constant.


The function f(x) is called the integrand and f(x) dx is known as the element of integration.


Some Elementary Standard Integrals in Integration

\[\int x^{n}dx\]

\[\frac{x^{n+1}}{n+1} + C\], where n \[\neq -1\]

\[\int sin x dx\]

- cos x + C

\[\int cos x dx\]

sin x + C

\[\int sec^{2} x dx\]

tan x + C

\[\int cosec^{2} x dx\]

- cot x + C

\[\int {\text{secx tanx dx}}\]

sec x + C

\[\int {\text{cosecx cotx dx}}\]

- cosec x + C


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 -


\[\int_{a}^{b}f(x)dx\]


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

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Definite Integral

(Limit goes from a to b where a is the lower limit and b is the upper limit)

Indefinite Integral

(Here there are no specific values, means there is no limit)


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

\[\int_{a}^{b}f(x)dx\] = F(b) - F(a)


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.


\[\int_{a}^{b}f(x)dx\] is equal to Area that lies above the x axis - Area that lies below the y axis.


  1. 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:


\[\int_{a}^{b}f(x) + g(x) dx\] is equal to \[\int_{a}^{b}f(x)dx\] + \[\int_{a}^{b}g(x)dx\]


  1. Reversing the Interval

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

\[\int_{a}^{b}f(x)dx\] is equal to - \[\int_{b}^{a}f(x)dx\]

 

  1. 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:


\[\int_{a}^{a}f(x)dx\] = 0


  1. Adding Intervals

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


\[\int_{a}^{b}f(x)dx\] = \[\int_{a}^{c}f(x)dx\] + \[\int_{b}^{c}f(x)dx\]


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

FAQ (Frequently Asked Questions)

Question 1) What does a Definite Integral Actually Calculate?

Answer) Evaluating a definite integral generally means to find the area enclosed by the graph of the function and the x-axis, over the given interval that is [a,b].


In the graph given below, the shaded area denotes the integral of f(x) on the given interval [a,b]. Now ,finding this area means that we are taking the integral of the function f(x), plugging the upper limit that is b into the result, and then subtracting from that whatever you get when you plug in the lower limit that is a.


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Question 2) Can Definite Integrals be Negative?What is a Definite Integral Used for?

Answer) Yes,we can say that a definite integral can be negative. Integrals basically measure the area between the curve in question and the x-axis over a specified interval. If all of the area within the interval exists below the x-axis yet above the curve then the result we get is negative.


Definite Integral. The definite integral can be defined to be exactly the limit and summation that we looked at to find the net area between a function and the x -axis. We should also keep in mind that the notation for the definite integral is very similar to the notation for an indefinite integral.