# Exercise 8.1(Application of Integrals)

### Chapter 8 Exercise 8.1 ncert math solution class 12

* Question1: *Find the area of the region bounded by the curve and the lines and the x-axis.

**Solution :** The area of the region bounded by the curve, , the lines, and , and the -axis is the area .

Area of

units

**Question 2:** Find the area of the region bounded by and the -axis in the first quadrant.

**Solution :** The area of the region bounded by the curve, , and , and the -axis, is the area .

Area of

units

**Question 3:** Find the area of the region bounded by and the -axis in the first quadrant.

**Solution :** The area of the region bounded by the curve, , and , and the -axis is the area .

Area of

units

**Question 4:** Find the area of the region bounded by the ellipse

**Solution :** The given equation of the ellipse

It can be observed that the ellipse is symmetrical about -axis and -axis.

Area bounded by ellipse Area of OAB

Area of

Therefore, area bounded by the ellipse Area of units

**Question 5:** Find the area of the region bounded by the ellipse

**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 Area of OAB

Therefore, area bounded by the ellipse Area of units

**Question 6:** Find the area of the region in the first quadrant enclosed by -axis, line and the

**Solution :** The area of the region bounded by the circle, ,line and the x-axis is the area OAB.

Solving and

and

The point of intersection of the line and the circle in the first quadrant is .

Area Area Area

Area of

….(i)

…..(ii)

Adding the equation (1) and (2), we get

The total area of units

Therefore, area enclosed by – axis, the line and the circle in the first quadrant is square units.

**Question 7:** Find the area of the smaller part of the circle cut off by the line .

**Solution :** It can be observed that the area is symmetrical about -axis.

Given,

Area of ABC

Therefore, the area of smaller part of the circle, , cut off by the line is square units.

**Question 8:** The area between and is divided into two equal parts by the line , find the value of a.

**Solution :** The line, , divides the area bounded by the parabola and into two equal parts.

Area Area

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

Area Area

Area

…..(i)

Area of

…..(ii)

From (1) and (2), we have

Therefore, the value of is .

**Question 9:** Find the area of the region bounded by the parabola and

**Solution:** The area bounded by the parabola, , and the line, , can be represented as The given area is symmetrical about y-axis.

Area OACO Area ODBO

Solving and

and and

The point of intersection of parabola, , and line, , is and .

Area of Area

Area of

Area of

Area of Area of

Therefore, required area units

**Question 10:** Find the area bounded by the curve and the line .

**Solution :** The area bounded by the curve and the line is represented by the shaded area OBAO.

Solving and

and

aand and

Let and be the points of intersection of the line and parabola. Coordinates of point are and point are .

We draw AL and BM perpendicular to x-axis.

It can be observed that,

Area OBAO

**Question 11:**Find the area of the region bounded by the curve and the line .

Solution : The region bounded by the parabola, , and the line, , is the area OABO.

The area OABO is symmetrical about x-axis.

Area of (Area of

Area

Therefore, the required area is units

**Question 12:** Area lying in the first quadrant and bounded by the circle and the lines and is

(A)

(B)

(C)

(D)

**Solution : The correct answer is (A).**

The area bounded by the circle and the lines, and , in the first quadrant is represented as

Area of

units

Thus, the correct answer is (A).

**Question 13:** Area of the region bounded by the curve -axis and the line is

(A) 2

(B)

(C)

(D)

**Solution : The correct answer is (B)**

The area bounded by the curve, -axis and is represented as

Area

units

Thus, the correct answer is (B).

## Chapter 8: Application of Integrals Class 12

Exercise 8.1 ncert math solution class 12