## NCERT Solutions for Class 12 Maths Chapter 8 – Application of Integrals Ex 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.

#### 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 = 9xx = 2, and x = 4, and the x-axis is the area ABCD.

#### Question 3:

Find the area of the region bounded by x2 = 4yy = 2, y = 4 and the y-axis in the first quadrant.

#### Answer:

The area of the region bounded by the curve, x2 = 4yy = 2, and y = 4, and the y-axis is the area ABCD.

#### 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
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
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.
The point of intersection of the line and the circle in the first quadrant is .
Area OAB = Area ΔOCA + Area ACB
Area of OAC
Area of ABC
Therefore, required area enclosed =
32 + π3 – 32 = π3 square units

#### Question 7:

Find the area of the smaller part of the circle x2 + y2 = a2 cut off by the line

#### Answer:

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
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.
âˆ´ Area OAD = Area ABCD
It can be observed that the given area is symmetrical about x-axis.
â‡’ Area OED = Area EFCD

From (1) and (2), we obtain
Therefore, the value of a is .

#### Question 9:

Find the area of the region bounded by the parabola x2 and

#### Answer:

The area bounded by the parabola, x2 = y,and the line,, can be represented as
The given area is symmetrical about y-axis.
âˆ´ Area OACO = Area ODBO
The point of intersection of parabola, x2 = y, and line, x, is A (1, 1).
Area of OACO = Area Î”OAM â€“ Area OMACO
Area of Î”OAM
Area of OMACO
â‡’ Area of OACO = Area of Î”OAM â€“ Area of OMACO
Therefore, required area = units

#### Question 10:

Find the area bounded by the curve x2 = 4y and the line x = 4– 2

#### Answer:

The area bounded by the curve, x2 = 4y, and line, x = 4– 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
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.
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 = 2 is
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.

#### Question 13:

Area of the region bounded by the curve y2 = 4xy-axis and the line y = 3 is
A. 2
B.
C.
D.

#### Answer:

The area bounded by the curve, y2 = 4xy-axis, and y = 3 is represented as
Thus, the correct answer is B.

Courtesy : CBSE