Examples on Applications of Integrals
Example1: Determine the area enclosed between the curve represented by y2 =4x and the linear function y=x.
Solution:
Given,
- x2 = 4x
Rearrange it to obtain a quadratic equation:
x2 β 4x = 0
x(xβ4) = 0
x = 0 and x = 4
Now, use the definite integral for the calculation of area between the curve and the line:
Area = β«04 (y β y2/2).dx
Here, y = x is the upper curve, and y2 = 4x is the lower curve.
Area = β«04 (x β x2/4).dx
Now, integrate with respect to x,
Area = [x2/2 β x3/12]04
Area = (42/2β43/12) β (02/2β03/12)
Area = (8β64/12) β (0β0)
Area = (8β16/3)
Area = 24/3 β 16/3 = 8/3
Therefore, the Area of the region enclosed between the curve y2 = 4x and the straight line y = x is 8/3 square units.
Example 2: Determine the area enclosed between the curve represented by y2 =4y-x and y axis.
Solution:
Write the curve equation in terms of y: y2 = 4yβx
y2 β 4y + x = 0
Put x in terms of y: x = y2 β 4y
Put x=0 and solve for y: 0 = y2 β 4y
y(yβ4) = 0
So, y = 0 and y = 4 are the points where the curve intersects the y-axis.
Area between the curve and the y-axis is given by the integral,
A = β«y1y2 x.dy
In this case, y1 = 0 and y2 = 4
A = β«04 (y2 β 4y).dy
A = [y3/3β2y2]04
A = (43/3β2(42)) β (03/3β2(02))
A = -32/3
Example 3: The area bounded by the curve y=3x2 and the lines x = -2, x = 3 and x-axis. Find the area under this curve.
Solution:
Curve is bounded by the lines x = β2 and x =3
Given Curve,
- y = 3x2
Area (A) under the curve and between the specified lines is given by the integral,
A = β«-23 y.dx
A = β«-23 3x2.dx
A = [x3]β23 = 35
Application of Integrals
Application of Integrals are used to find the area and volume of various 2-D and 3-D curves and they have vast applications in the fields of mathematics and physics. They generally help us to calculate the area of the curve, irregular contour, the volume of various curves, and others.
In this article, we will discuss Applications of Integral and its meaning. We will also understand its application and formulae to calculate the Integral value in mathematics. We will also solve various examples and provide practice questions for a better understanding of the concept of this article. We have to study the Application of Integrals in Class 12.
Table of Content
- Application of Integral in Mathematics
- Types of Integrals
- Application of Integrals
- Application of Integrals in Physics
- Examples on Applications of Integrals
Contact Us