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

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

Application of Integrals Important Questions

Application of Integrals Objective Type Questions

Question 1.
Choose the correct answer:

Question 1.
The value of \(\int_{0}^{\frac{\pi}{2}} \frac{\sin x}{\sin x+\cos x} d x\) is:
(a) π
(b) \(\frac { \pi }{ 2 } \)
(c) \(\frac { \pi }{ 4 } \)
(d) \(\frac { -\pi }{ 4 } \)
Answer:
(c) \(\frac { \pi }{ 4 } \)

Question 2.
Area of the best quadrant of the ellipse \(\frac { x^{ 2 } }{ a^{ 2 } } \) + \(\frac { y^{ 2 } }{ b^{ 2 } } \) = 1 is:
(a) πab
(b) \(\frac{1}{2}\) πab
(c) \(\frac{1}{4}\) πab
(d) \(\frac{1}{8}\) πab
Answer:
(c) \(\frac{1}{4}\) πab

MP Board Solutions

Question 3.
The value of \(\int_{2}^{4} x^{3} d x\) is:
(a) 60
(b) 50
(c) 70
(d) 256
Answer:
(a) 60

Question 4.
\(\int_{0}^{2 a} f(x) d x=0\), if:
(a) f(2a – x) = f(x)
(b) f(2a – x) = – f(x)
(c) f(x) is an even function
(d) f(x) is an odd function
Answer:
(b) f(2a – x) = – f(x)

Question 5.
The value of \(\int_{0}^{2 \pi}|\sin x| d x\) is equal to:
(a) 2
(b) \(\sqrt{3}\)
(c) 4
(d) 0
Answer:
(d) 0

Question 2.
Fill in the blanks:
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals
Answer:

  1. 0
  2. 0
  3. 17
  4. 0
  5. \(\frac { \pi }{ 4 } \)
  6. πa2

Question 3.
Write True/False:
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals
Answer:

  1. True
  2. True
  3. False
  4. False
  5. True
  6. True

Question 4.
Match the column:
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals
Answer:

  1. (b)
  2. (a)
  3. (d)
  4. (c)
  5. (f)
  6. (e)

Question 5.
Write the answer in one word/sentence:
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals
Answer:

  1. 1
  2. \(\frac { -\pi }{ 2 } \) log 2
  3. 2
  4. \(\frac { \pi }{ 2 } \)
  5. -4
  6. \(\frac { \pi }{ 12 } \)

Application of Integrals Long Answer Type Questions – II

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),
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals
Coordinate of A(8, 8)
Two parabolas intersect at point O and A.
∴ Required area = Area OBALO – Area OCALO
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

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
When x ∈ [0, \(\frac { \pi }{ 2 } \), cos x ≥ 0;
x ∈ [0, \(\frac { \pi }{ 2 } \), [0, \(\frac { 3\pi }{ 2 } \), cos x ≤ 0;
When x ∈ \(\frac { 3\pi }{ 2 } \), 2π] cos x ≥ 0
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

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

MP Board Solutions

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)
⇒ \(\frac { x }{ a } \) + \(\frac { y }{ a } \) = 1, \(\frac { x }{ -a } \) + \(\frac { y }{ -a } \) = 1
⇒ \(\frac { x }{ a } \) + \(\frac { y }{ -a } \) = 1, \(\frac { x }{ -a } \) + \(\frac { y }{ a } \) = 1
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals
The above graph is represented by the PQ, RS, PS and QR respectively.
Hence, the area bounded by these lines
= 4 × ∆Area of OPQ
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

Question 5.
(A) Find the rea of circle x2 + y2 = a2 by using integration method?
Solution:
Given:
x2 + y2 = a2
⇒ y2 = a2 – x2
⇒ y = \(\sqrt { a^{ 2 }-x^{ 2 } } \)
The circle is symmetric about the axes
Area = 4 × area of quadrant OAB
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

(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
Putting the value of y from eqn. (2) in eqn. (1), we have
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals
From eqn.(2), we have
y = 0, y = \(\frac { (4a)^{ 2 } }{ 4a } \) = 4a
∴The points of intersection of the given parabolas are (0, 0) and (4a, 4a).
Now, Required area = Area OBAL – Area OCAL
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

MP Board Solutions

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
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals
∴ At O, x = 0 and at M, x = 4
Hence Required area Area OAPB O
= Area OAPMO – Area OBPMO
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

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 \(\frac { x^{ 2 } }{ a^{ 2 } } \) + \(\frac { y^{ 2 } }{ b^{ 2 } } \) = 1?
Solution:
Equation of the ellipse is
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals
∴ Required area = 4 area AOB
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

Question 10.
Find the area of the region bounded by the parabola y2 = 4ax and its latus rectum?
Solution:
Equation of the parabola is
y2 = 4ax or y = ±2 \(\sqrt { ax } \)
In the figure, LSL’ is latus rectum and vertex is O (0, 0)
∴ Required area = 2 × Area OSL
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

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
∴ x = 0, x = \(\frac { 4a }{ m^{ 2 } } \)
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

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
⇒ x = \(2\sqrt { y } \)
Required area = Area of ABCD
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

Question 13.
Find the area of the smaller part of the circle x2 + y2 = a2 cut off by the line x = \(\frac { a }{ \sqrt { 2 } } \)? (NCERT)
Solution:
Given equation of curve is:
x2 + y2 = a2 …………………………. (1)
⇒ y2 = a2 – x2
⇒ y = \(\sqrt { a^{ 2 }-x^{ 2 } } \)
Equation of line is:
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals
Point of intersection of circle and line is A \(\frac { a }{ \sqrt { 2 } } \), \(\frac { a }{ \sqrt { 2 } } \)
Coordinates of D is (a, 0).
∴ Required area = Area of ABD
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

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 = \(\sqrt{x}\)
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
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

MP Board Solutions

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
⇒ y = \(\sqrt { (2)^{ 2 }-x^{ 2 } } \)
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

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:
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals
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 (4\(\sqrt{2}\),0).
Required area = Area OACBO
= Area OAB + Area ABC
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals
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
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals
MP Board Class 12th Maths Important Questions Chapter 8 Application of Integrals

MP Board Class 12 Maths Important Questions