Chapter 8 application_of_integrals

Class XII Chapter 8 – Application of Integrals Maths
Page 1 of 53
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717
Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051
(One Km from ‘Welcome’ Metro Station)
Exercise 8.1
Question 1:
Find the area of the region bounded by the curve y2
= x and the lines x = 1, x = 4 and
the x-axis.
Answer
The area of the region bounded by the curve, y2
= x, the lines, x = 1 and x = 4, and the
x-axis is the area ABCD.

Page 2 of 53
Question 2:
Find the area of the region bounded by y2
= 9x, x = 2, x = 4 and the x-axis in the first
quadrant.
Answer
The area of the region bounded by the curve, y2
= 9x, x = 2, and x = 4, and the x-axis
is the area ABCD.

Page 3 of 53
Question 3:
Find the area of the region bounded by x2
= 4y, y = 2, y = 4 and the y-axis in the first
quadrant.
Answer
The area of the region bounded by the curve, x2
= 4y, y = 2, and y = 4, and the y-axis
is the area ABCD.

Page 4 of 53
Question 4:
Find the area of the region bounded by the ellipse
Answer
The given equation of the ellipse, , can be represented as
It can be observed that the ellipse is symmetrical about x-axis and y-axis.
∴ Area bounded by ellipse = 4 × Area of OAB

Page 5 of 53
Therefore, area bounded by the ellipse = 4 × 3π = 12π units
Question 5:
Find the area of the region bounded by the ellipse
Answer
The given equation of the ellipse can be represented as
It can be observed that the ellipse is symmetrical about x-axis and y-axis.
∴ Area bounded by ellipse = 4 × Area OAB

Page 6 of 53
Therefore, area bounded by the ellipse =
Question 6:
Find the area of the region in the first quadrant enclosed by x-axis, line and the
circle
Answer
The area of the region bounded by the circle, , and the x-axis is the
area OAB.

Page 7 of 53
The point of intersection of the line and the circle in the first quadrant is .
Area OAB = Area ∆OCA + Area ACB
Area of OAC
Area of ABC
Therefore, area enclosed by x-axis, the line , and the circle in the first
quadrant =
Question 7:
Find the area of the smaller part of the circle x2
+ y2
= a2
cut off by the line
Answer

Page 8 of 53
The area of the smaller part of the circle, x2
+ y2
= a2
, cut off by the line, , is the
area ABCDA.
It can be observed that the area ABCD is symmetrical about x-axis.
∴ Area ABCD = 2 × Area ABC

Page 9 of 53
Therefore, the area of smaller part of the circle, x2
+ y2
= a2
, cut off by the line, ,
is units.
Question 8:
The area between x = y2
and x = 4 is divided into two equal parts by the line x = a, find
the value of a.
Answer
The line, x = a, divides the area bounded by the parabola and x = 4 into two equal
parts.

Page 10 of 53
∴ Area OAD = Area ABCD
It can be observed that the given area is symmetrical about x-axis.
⇒ Area OED = Area EFCD

Page 11 of 53
From (1) and (2), we obtain
Therefore, the value of a is .
Question 9:
Find the area of the region bounded by the parabola y = x2
and
Answer
The area bounded by the parabola, x2
= y,and the line, , can be represented as

Page 12 of 53
The given area is symmetrical about y-axis.
∴ Area OACO = Area ODBO
The point of intersection of parabola, x2
= y, and line, y = x, is A (1, 1).
Area of OACO = Area ∆OAB – Area OBACO
⇒ Area of OACO = Area of ∆OAB – Area of OBACO
Therefore, required area = units

Page 13 of 53
Question 10:
Find the area bounded by the curve x2
= 4y and the line x = 4y – 2
Answer
The area bounded by the curve, x2
= 4y, and line, x = 4y – 2, is represented by the
shaded area OBAO.
Let A and B be the points of intersection of the line and parabola.
Coordinates of point .
Coordinates of point B are (2, 1).
We draw AL and BM perpendicular to x-axis.
It can be observed that,
Area OBAO = Area OBCO + Area OACO … (1)
Then, Area OBCO = Area OMBC – Area OMBO

Page 14 of 53
Similarly, Area OACO = Area OLAC – Area OLAO
Therefore, required area =
Question 11:
Find the area of the region bounded by the curve y2
= 4x and the line x = 3
Answer
The region bounded by the parabola, y2
= 4x, and the line, x = 3, is the area OACO.

Page 15 of 53
The area OACO is symmetrical about x-axis.
∴ Area of OACO = 2 (Area of OAB)
Therefore, the required area is units.
Question 12:
Area lying in the first quadrant and bounded by the circle x2
+ y2
= 4 and the lines x = 0
and x = 2 is

Page 16 of 53
A. π
B.
C.
D.
Answer
The area bounded by the circle and the lines, x = 0 and x = 2, in the first quadrant is
represented as
Thus, the correct answer is A.

Page 17 of 53
Question 13:
Area of the region bounded by the curve y2
= 4x, y-axis and the line y = 3 is
A. 2
B.
C.
D.
Answer
The area bounded by the curve, y2
= 4x, y-axis, and y = 3 is represented as
Thus, the correct answer is B.

Page 18 of 53
Exercise 8.2
Question 1:
Find the area of the circle 4x2
+ 4y2
= 9 which is interior to the parabola x2
= 4y
Answer
The required area is represented by the shaded area OBCDO.
Solving the given equation of circle, 4x2
+ 4y2
= 9, and parabola, x2
= 4y, we obtain the
point of intersection as .
It can be observed that the required area is symmetrical about y-axis.
∴ Area OBCDO = 2 × Area OBCO
We draw BM perpendicular to OA.
Therefore, the coordinates of M are .
Therefore, Area OBCO = Area OMBCO – Area OMBO

Page 19 of 53
Therefore, the required area OBCDO is
units
Question 2:
Find the area bounded by curves (x – 1)2
+ y2
= 1 and x2
+ y 2
= 1
Answer
The area bounded by the curves, (x – 1)2
+ y2
= 1 and x2
+ y 2
= 1, is represented by
the shaded area as

Page 20 of 53
On solving the equations, (x – 1)2
+ y2
= 1 and x2
+ y 2
= 1, we obtain the point of
intersection as A and B .
It can be observed that the required area is symmetrical about x-axis.
∴ Area OBCAO = 2 × Area OCAO
We join AB, which intersects OC at M, such that AM is perpendicular to OC.
The coordinates of M are .

Page 21 of 53
Therefore, required area OBCAO = units
Question 3:
Find the area of the region bounded by the curves y = x2
+ 2, y = x, x = 0 and x = 3
Answer
The area bounded by the curves, y = x2
+ 2, y = x, x = 0, and x = 3, is represented by
the shaded area OCBAO as

Page 22 of 53
Then, Area OCBAO = Area ODBAO – Area ODCO
Question 4:
Using integration finds the area of the region bounded by the triangle whose vertices are
(–1, 0), (1, 3) and (3, 2).
Answer
BL and CM are drawn perpendicular to x-axis.
It can be observed in the following figure that,
Area (∆ACB) = Area (ALBA) + Area (BLMCB) – Area (AMCA) … (1)

Page 23 of 53
Equation of line segment AB is
Equation of line segment BC is
Equation of line segment AC is
Therefore, from equation (1), we obtain

Page 24 of 53
Area (∆ABC) = (3 + 5 – 4) = 4 units
Question 5:
Using integration find the area of the triangular region whose sides have the equations y
= 2x +1, y = 3x + 1 and x = 4.
Answer
The equations of sides of the triangle are y = 2x +1, y = 3x + 1, and x = 4.
On solving these equations, we obtain the vertices of triangle as A(0, 1), B(4, 13), and C
(4, 9).
Area (∆ACB) = Area (OLBAO) –Area (OLCAO)
Question 6:
Smaller area enclosed by the circle x2
+ y2
= 4 and the line x + y = 2 is
A. 2 (π – 2)

Page 25 of 53
B. π – 2
C. 2π – 1
D. 2 (π + 2)
Answer
The smaller area enclosed by the circle, x2
+ y2
= 4, and the line, x + y = 2, is
represented by the shaded area ACBA as
Area ACBA = Area OACBO – Area (∆OAB)
Question 7:
Area lying between the curve y2
= 4x and y = 2x is
A.

Page 26 of 53
B.
C.
D.
Answer
The area lying between the curve, y2
= 4x and y = 2x, is represented by the shaded
area OBAO as
The points of intersection of these curves are O (0, 0) and A (1, 2).
We draw AC perpendicular to x-axis such that the coordinates of C are (1, 0).
∴ Area OBAO = Area (∆OCA) – Area (OCABO)

Page 27 of 53

Page 28 of 53
Miscellaneous Solutions
Question 1:
Find the area under the given curves and given lines:
(i) y = x2
, x = 1, x = 2 and x-axis
(ii) y = x4
, x = 1, x = 5 and x –axis
Answer
i. The required area is represented by the shaded area ADCBA as
ii. The required area is represented by the shaded area ADCBA as

Page 29 of 53
Question 2:
Find the area between the curves y = x and y = x2
Answer
The required area is represented by the shaded area OBAO as

Page 30 of 53
The points of intersection of the curves, y = x and y = x2
, is A (1, 1).
We draw AC perpendicular to x-axis.
∴ Area (OBAO) = Area (∆OCA) – Area (OCABO) … (1)
Question 3:
Find the area of the region lying in the first quadrant and bounded by y = 4x2
, x = 0, y
= 1 and y = 4

Page 31 of 53
Answer
The area in the first quadrant bounded by y = 4x2
, x = 0, y = 1, and y = 4 is
represented by the shaded area ABCDA as
Question 4:
Sketch the graph of and evaluate
Answer

Page 32 of 53
The given equation is
The corresponding values of x and y are given in the following table.
x – 6 – 5 – 4 – 3 – 2 – 1 0
y 3 2 1 0 1 2 3
On plotting these points, we obtain the graph of as follows.
It is known that,

Page 33 of 53
Question 5:
Find the area bounded by the curve y = sin x between x = 0 and x = 2π
Answer
The graph of y = sin x can be drawn as
∴ Required area = Area OABO + Area BCDB
Question 6:
Find the area enclosed between the parabola y2
= 4ax and the line y = mx
Answer
The area enclosed between the parabola, y2
= 4ax, and the line, y = mx, is represented
by the shaded area OABO as

Page 34 of 53
The points of intersection of both the curves are (0, 0) and .
We draw AC perpendicular to x-axis.
∴ Area OABO = Area OCABO – Area (∆OCA)

Page 35 of 53
Question 7:
Find the area enclosed by the parabola 4y = 3x2
and the line 2y = 3x + 12
Answer
The area enclosed between the parabola, 4y = 3x2
, and the line, 2y = 3x + 12, is
represented by the shaded area OBAO as
The points of intersection of the given curves are A (–2, 3) and (4, 12).
We draw AC and BD perpendicular to x-axis.
∴ Area OBAO = Area CDBA – (Area ODBO + Area OACO)

Page 36 of 53
Question 8:
Find the area of the smaller region bounded by the ellipse and the line
Answer
The area of the smaller region bounded by the ellipse, , and the line,
, is represented by the shaded region BCAB as

Page 37 of 53
∴ Area BCAB = Area (OBCAO) – Area (OBAO)

Page 38 of 53
Question 9:
Find the area of the smaller region bounded by the ellipse and the line
Answer
The area of the smaller region bounded by the ellipse, , and the line,
, is represented by the shaded region BCAB as
∴ Area BCAB = Area (OBCAO) – Area (OBAO)

Page 39 of 53
Question 10:
Find the area of the region enclosed by the parabola x2
= y, the line y = x + 2 and x-
axis
Answer
The area of the region enclosed by the parabola, x2
= y, the line, y = x + 2, and x-axis
is represented by the shaded region OABCO as

Page 40 of 53
The point of intersection of the parabola, x2
= y, and the line, y = x + 2, is A (–1, 1).
∴ Area OABCO = Area (BCA) + Area COAC

Page 41 of 53
Question 11:
Using the method of integration find the area bounded by the curve
[Hint: the required region is bounded by lines x + y = 1, x – y = 1, – x + y = 1 and – x
– y = 11]
Answer
The area bounded by the curve, , is represented by the shaded region ADCB
as
The curve intersects the axes at points A (0, 1), B (1, 0), C (0, –1), and D (–1, 0).
It can be observed that the given curve is symmetrical about x-axis and y-axis.
∴ Area ADCB = 4 × Area OBAO

Page 42 of 53
Question 12:
Find the area bounded by curves
Answer
The area bounded by the curves, , is represented by the
shaded region as
It can be observed that the required area is symmetrical about y-axis.
Question 13:
Using the method of integration find the area of the triangle ABC, coordinates of whose
vertices are A (2, 0), B (4, 5) and C (6, 3)

Page 43 of 53
Answer
The vertices of ∆ABC are A (2, 0), B (4, 5), and C (6, 3).
Equation of line segment AB is
Equation of line segment BC is
Equation of line segment CA is

Page 44 of 53
Area (∆ABC) = Area (ABLA) + Area (BLMCB) – Area (ACMA)
Question 14:
Using the method of integration find the area of the region bounded by lines:
2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0
Answer
The given equations of lines are
2x + y = 4 … (1)
3x – 2y = 6 … (2)
And, x – 3y + 5 = 0 … (3)

Page 45 of 53
The area of the region bounded by the lines is the area of ∆ABC. AL and CM are the
perpendiculars on x-axis.
Area (∆ABC) = Area (ALMCA) – Area (ALB) – Area (CMB)
Question 15:
Find the area of the region
Answer
The area bounded by the curves, , is represented as

Page 46 of 53
The points of intersection of both the curves are .
The required area is given by OABCO.
It can be observed that area OABCO is symmetrical about x-axis.
∴ Area OABCO = 2 × Area OBC
Area OBCO = Area OMC + Area MBC
Question 16:
Area bounded by the curve y = x3
, the x-axis and the ordinates x = –2 and x = 1 is
A. – 9
B.

Page 47 of 53
C.
D.
Answer
Question 17:
The area bounded by the curve , x-axis and the ordinates x = –1 and x = 1 is
given by
[Hint: y = x2
if x > 0 and y = –x2
if x < 0]

Page 48 of 53
A. 0
B.
C.
D.
Answer
Thus, the correct answer is C.

Page 49 of 53
Question 18:
The area of the circle x2
+ y2
= 16 exterior to the parabola y2
= 6x is
A.
B.
C.
D.
Answer
The given equations are
x2
+ y2
= 16 … (1)
y2
= 6x … (2)
Area bounded by the circle and parabola

Page 50 of 53
Area of circle = π (r)2
= π (4)2
= 16π units
Thus, the correct answer is C.

Page 51 of 53
Question 19:
The area bounded by the y-axis, y = cos x and y = sin x when
A.
B.
C.
D.
Answer
The given equations are
y = cos x … (1)
And, y = sin x … (2)
Required area = Area (ABLA) + area (OBLO)
Integrating by parts, we obtain

Page 52 of 53

Page 53 of 53
Therefore, the required area is units

Chapter 8 application_of_integrals

More Related Content

What's hot

Similar to Chapter 8 application_of_integrals

More from Shahrukh Javed

Recently uploaded

Chapter 8 application_of_integrals