MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

October 16, 2025October 16, 2025

Important Questions

Long Answer Type Questions

Question 1.
Find the area bounded by the parabolas x² = 8y and y² = 8x?
Solution:
Equation of given parabolas:
x² = 8y ………………. (1)
and y² = 8x ………………….. (2)
Solving eqns. (1) and (2),

Coordinate of A(8, 8)
Two parabolas intersect at point O and A.
∴ Required area = Area OBALO – Area OCALO

Question 2.
Find the area bounded by the curve y = cos x, the X – axis and x = 0, x = 2π, by integration method?
Solution:
y = f(x) = cos x

Question 3.
(A) Find the area between the curves y² = 4x and y = 2x? (Integral method)
Solution:
Given, Curve and line are y² = 4x and – y = 2x
(2x)² = 4x
⇒ 4x² = 4x
⇒ x² – x = 0
⇒ x (x – 1) = 0
∴ x = 0 or x = 1
Then y = 2x = 0 or y = 2 × 1 = 2

The point of intersections of curve and line are (0, 0) and (1,2).
Area of shaded region = ar (OPAMO) – ar (OQAMO)

MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

(B) Find the area bounded by the curves x² = 4y and x = 4y – 2 by integration method?
Solution:
Given, Curve and line are y² = 4x and – y = 2x
(2x)² = 4x
⇒ 4x² = 4x
⇒ x² – x = 0
⇒ x (x – 1) = 0
∴ x = 0 or x = 1
Then y = 2x = 0 or y = 2 × 1 = 2

The point of intersections of curve and line are (0, 0) and (1,2).
Area of shaded region = ar (OPAMO) – ar (OQAMO)

MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

Question 4.
Find the area bounded by lines |x|+|y|= a by using integration method?
Solution:
The given lines are represented by |x|+|y| = a will be
x + y = a ……………. (1)
-x – y = a ……………………. (2)
x – y = a ……………………. (3)
and – x + y = a ………………… (4)

The above graph is represented by the PQ, RS, PS and QR respectively.
Hence, the area bounded by these lines
= 4 × ∆Area of OPQ

Question 5.
(A) Find the rea of circle x² + y² = a² by using integration method?
Solution:
Given:
x² + y² = a²
⇒ y² = a² – x²

The circle is symmetric about the axes
Area = 4 × area of quadrant OAB

(B)
Find the area of the circle x² + y² = 25 by using integration method?
Solution:
Solve as Q. No. 5 (A) by putting a = 5.

Question 6.
Find the area included between two parabolas:
y² = 4 ax and x² = 4ay, a > 0.
Solution:
The equations of the given parabolas are
y² = 4ax.
and x² = 4ay

∴The points of intersection of the given parabolas are (0, 0) and (4a, 4a).
Now, Required area = Area OBAL – Area OCAL

Question 7.
Find the area enclosed between parabolas y² = 4x and x² = 4y?
Solution:
y² = 4x ……………… (1)
x² = 4y …………………. (2)
Putting the value of y in eqn.(1), we get

∴ At O, x = 0 and at M, x = 4
Hence Required area Area OAPB O
= Area OAPMO – Area OBPMO

Question 8.
Find the area between curve y² = x and x² = y by integral method?
Solution:
Solve as Q.No. 7.

Question 9.
Find the area enclosed by the ellipse ?
Solution:

∴ Required area = 4 area AOB

Question 10.
Find the area of the region bounded by the parabola y² = 4ax and its latus rectum?
Solution:
Equation of the parabola is

Question 11.
Find the area enclosed between parabola y² = 4ax and straight line y = mx?
Solution:
Equation of the parabola y² = 4ax ……………. (1)
Equation of straight line y = mx ………………….. (2)
O is the origin (O, O). P is the point of intersection of parabola and the straight line. Solving eqns. (1) and (2) we get,
y² = 4ax
⇒ (mx)² = 4ax
⇒ m²x² – 4ax = 0
⇒ x(m² – 4a) = 0

Question 12.
Find the area bounded by the curve x² = 4y, y = 2, y = 4 and Y – axis in the first quadrant? (NCERT)
Solution:
Given equation of curve is:
x² = 4y

Question 13.
Find the area of the smaller part of the circle x² + y² = a² cut off by the line x = ? (NCERT)
Solution:
Given equation of curve is:
x² + y² = a² …………………………. (1)
⇒ y² = a² – x²

MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

Question 14.
The area between x = y² and x = 4 is divided into two equal parts by the line x = a. Find the value of a? (NCERT)
Solution:
Given:
Equation of parabola is
x = y²
⇒ y = √x
Area bounded by the parabola and the line x = 4 is divided into two equal parts by the line x = a.
Area OEC = Area EFCB

Question 15.
Find the area lying in the first quadrant and bounded by the circle x² + y² = 4 and the lines x = 0 and x = 2? (NCERT)
Solution:
Given equation of circle is:
x² + y² = 4
⇒ y² = 4 – x²

Question 16.
Find the area enclosed by a circle x² + y² = 32, a line y = x and axis of A in the first quadrant?
Solution:
Equation of circle is:

Putting the value of x in eqn. (2)
y = 4
The point of intersection of the line and circle are 0(0,0) and A(4,4) coordinates of B are are (4,0) and coordinates of C are (42–√,0).
Required area = Area OACBO
= Area OAB + Area ABC

MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

Question 17.
Find the area of triangle whose sides are y = 2x + 1, y = 3x +1 and x = 4? (NCERT)
Solution:
Let the equation of sides AB,AC and BC of & ABC ore respectively.
y = 2x + 1 …………… (1)
y = 3x + 1 ………….. (2)
and x = 4 …………….. (3)
By solving eqns. (1) and (2) we have A (0, 1), by solving eqns. (1) and (3) we have B (4, 9) and by solving eqns. (2) and (3) we have C (4, 13).
Thus required area