NCERT Solutions for Chapter 8 Application of Integrals Class 12 Maths Exercise 8.1

Class 12 Maths NCERT Solutions for Chapter 8 Application of Integrals Exercise 8.1

Integrals Exercise 8.1 Solutions

1. Find the area of the region bounded by the curve y² = x and the lines x = 1, x = 4 and the x - axis. 

Solution

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. 

Area of ABCD = ∫₁⁴ y dx 
=  ∫₁⁴ √x dx 

= (2/3) [8 - 1] 
= 14/3 units 

2. Find the area of the region bounded by y² = 9x, x = 2, x = 4 and the x-axis in the first quadrant.

Solution

The area of the region bounded by the curve, y² = 9x, x = 2, and x = 4, and the x - axis is the area ABCD.

Area of ABCD = ∫₂⁴ y dx
= ∫₂⁴ 3√x dx 

= 2[8 - 2√2]
= (16 - 4√2) units 

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

Solution

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

Area of ABCD = ∫₂⁴ y dx
= ∫₂⁴ 2√y dy
= 2∫₂⁴ √y dy

= (4/3)[8 - 2√2]
= (32 - 8√2)/3  units 

4. Find the area of the region bounded by the ellipse x²/16 + y²/9 = 1 

Solution

The given equation of the ellipse, x² /16 + y² /9 = 1 , 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 

Area of OAB = ∫₂⁴ y dx

Therefore, area bounded by the ellipse = 4 × 3π = 12π units 

5. Find the area of the region bounded by the ellipse  x²/4 + y²/9 = 1 

Solution

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 = 4× (3π/2) = 6π units 

6. Find the area of the region in the first quadrant enclosed by x-axis, line x = √3 y  and the circle x² + y² = 4. 

Solution

The area of the region bounded by the circle, x² + y² = 4, x = √3 y, and the x - axis is the area OAB. 

The point of intersection of the line and the circle in the first quadrant is (√3, 1) . 
Area OAB = Area ΔOCA + Area ACB 

Area of OAC = (1/2) × OC × AC = (1/2) × √3 × 1 = √3/2  ...(1)

Therefore, area enclosed by x - axis, the line x = √3 y, and the circle x² + y² = 4 in the 

first quadrant = √3/2  + π/2 - √3/2 =  π/3 units 

7. Find the area of the smaller part of the circle x² + y² = a² cut off by the line x = a/√2

Solution

The area of the smaller part of the circle, x² + y² = a² , cut off by the line, x = a/√2, is the area ABCD. 

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, x = a/√2, is [(a² /2)(π/2 - 1)] units. 

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.

Solution

The line, x = a, divides the area bounded by the parabola and x = 4 into two equal parts. 
Area OAD = Area ABCD 

It can be observed that the given area is symmetrical about x - axis. 

∴ Area OED  = Area EFCD 

From (1) and (2), we obtain 

⇒ 2.(a)^3/2 = 8 

⇒ (a)^3/2 = 4 
⇒ a = (4)^2/3 

Therefore, the value of a is (4)^2/3 . 

9. Find the area of the region bounded by the parabola y = x² and y = |x| .

Solution

The area bounded by the parabola, x² = y , and the line, y = |x|, can be represented as 

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).

Therefore, required area = 2[1/6] = 1/3 sq units 

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

Solution

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  A are (-1, 1/4). 

Coordinates of point B are (2, 1)
We draw AL and BM perpendicular to x - axis. 
Similarly, Area OACO = Area OLAC - Area OLAO 

Therefore, required area = (5/6 + 7/24) = 9/8 units 

11. Find the area of the region bounded by the curve y² = 4x and the line x = 3

Solution

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 8√3 units. 

12. Area lying in the first quadrant and bounded by the circle x² + y² = 4 and the lines x = 0 and x = 2 is 
(A) π 
(B) π/2
(C) π/3
(D) π/4

Solution

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. 

13. Area of the region bounded by the curve y² = 4x , y - axis and the line y = 3 is 

(A) 2
(B) 9/4 
(C) 9/3 
(D) 9/2 

Solution

The area bounded by the curve, y² = 4x, y - axis, and y = 3  is represented as 

Thus, the correct answer is B.