What is Integral? Explained with its types, applications, and graphs

 What is Integral? Explained with its types, applications, and graphs

Integrals are used in many domains, including mathematics, science, and engineering. We mostly utilize integral formulae to calculate areas.

So, let us offer a quick introduction to integrals based on the Mathematics topic in order to determine areas under simple curves, areas limited by a curve and a line, and areas between two curves, as well as the application of integrals in the mathematical disciplines and the solved problem.

What is an Integral?

Integral is the reverse of derivative. The answer of the integral is the function whose derivative is given. The process of finding the integral is known as integration. An integral is frequently used to evaluate the area under the curve.

Integration is also known as the anti-derivative of the function.

The integral formulae

The formulae of the integral are used to integrate the given function with or without limits.

The notation of integral is

                                                          ʃ f(x) dx

        f(x) is the function.

        dx is the integral variable of integration with respect we take the integral.

        ʃ the sign of integral.

Types of Integral

The integral (anti-derivative) has two types.

        Indefinite integral

        definite integral

We discuss this one by one in detail with examples.

        Indefinite integral

The indefinite integral is an integral of a function without using the upper and lower limits. it is represented by

ʃ f(x) dx = F(x) +C

        F(x) is the function after taking integral

        C is called the constant of integration

        f(x) is the function whose integral we have to find

        dx is called the integral variable

        Definite integral

The definite integral is another type of integral. In this integral lower limit and upper limit involves the integral sign.

ʃba f(x) dx = F(b) – F(a)

        f(x) is  a function.

         a and b is called the lower limit and upper limit of the function respectively.

        dx is known as the integral variable with respect to the procedure of integration carry.

        No constant involves

        We put the upper limit first and then minus the lower limit.

Examples of integral calculus

We discuss the each type with the help of examples and detailed solutions with step-by-step elaboration.

Example 1:

Evaluate the definite integral of f(x) = sin(5x) + x2 – 5x where x is an integrating variable and the variation of x is from 2 to 3.


Step 1: Change into the integral notation.

ʃ32 sin (5x) + x2 – 5x) dx

Step 2: Apply the integral separately on each term.

= ʃ32 [sin (5x)] dx + ʃ32 [x2] dx – ʃ32 [5x] dx

Step 3: Taking the coefficient outside of the integral sign.

= ʃ32 [sin (5x)] dx + ʃ32 [x2] dx – 5ʃ32 [5x] dx

Step 4: Now apply the integral

= [-cos (5x)/5]32 + [x2+1 / 2 + 1]32 – 5 [x1+1 /1 + 1]32

= [-cos (5x)/5]32 + [x3 / 3]32 – 5 [x2 /2]32

= -1/5[cos (5x)]32 + 1/3 [x3]32 – 5/2 [x2]32

Step 6: Put the limits according to the fundamental theorem of calculus.

= -1/5[cos (5(3)) – cos (5(2))] + 1/3 [33 – 23] – 5/2 [32 – 22]

= -1/5[cos (15) – cos (10)] + 1/3 [27 – 8] – 5/2 [9 – 4]

= -1/5[cos (15) – cos (10)] + 1/3 [19] – 5/2 [5]

= -1/5[cos (15) – cos (10)] + 19/3 – 25/2

= -1/5[cos (15) – cos (10)] + 37/6

= -6.1825

The calculation of the integral is sometimes a very difficult and time taking procedure. To avoid these difficulties try online integral calculators.

Solved through integral calculator by Meracalculator

Example 2:

 Find the area under the given function (16 x2).


Step 1: Find the limits

 (16 x2) = 0

Taking the square on both sides

 16 – x2 = 0

42 – x2 = 0

(4-x) (4 + x) = 0

4 - x = 0

4+x = 0

x = 4

x = -4

So the limits of integration are 4, -4

Step 2: Now write in the form of the definite integral.

ʃ4-4(16 x2)dx

Step 3: Draw the graph of the given function.

Step 4: The graph of the integrated function with limits -4 to 4 as the semi-circle with a radius of 4. Use the formula of the area of the semi-circle

Area of semi-circle = (pi*r^2)/2

Step 5: Put the value of the radius.

Area of the semi-circle = (3.14*42)/2

Area of the semi-circle = 25.12

Example 3:

Evaluate the indefinite integral of the function 2x3 – 9x2 + 5cos(10x) with integrate variable x.


Step 1: Change into the integral notation.

 ʃ [2x3- 9x2+5cos (10x)] dx

Step 2: Apply the integral according to the rule that it separates on each term

= ʃ (2x3) dx – ʃ (9x2) dx + ʃ 5cos (10x)) dx

Step 3: Take the coefficient outside the integral sign.

= 2ʃ (x3) dx – 9ʃ (x2) dx + 5ʃ cos (10x)) dx

Step 4: Apply integral now

= 2 (x3+1/ 3+1) – 9 (x2+1/ 2+1) + 5[sin (10x)/10] + C

= 2 (x4/4) – 9 (x3/ 3) + 5[sin (10x)/10] + C

= 2/4 (x4) – 9/3 (x3) + 5/10[sin (10x)] + C

= 1/2 (x4) – 3 (x3) + 1/2[sin (10x)] + C

= x4/2 – 3x3 + sin(10x)/2 + C

Application of integral in real life

The application of integrations in real life is determined by the industries in which this calculus is employed. We discuss some real-life applications the integral is following

        Engineers, for example, utilize integrals to determine the geometry of building projects or the length of electrical wire necessary to connect two substations.

        It is used in science to address questions about the center of gravity and other physics concerns.

         In the field of graphical representation, three-dimensional models are displayed.

        Kinetic energy and improper integrals

        Probability theory

        The area between the curve


In this post, we have learned about integration, its types, examples, and real-life applications.

You can now able to solve both types of integral questions easily and with better accuracy. The integral is one of the main branches of calculus. 

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