Chapter 12. Application of integrals

Class 12 Maths Chapter 12 Application of integrals Example Questions and Solutions

Example 1: Find the area enclosed by the circle  .

Solution:  The equation of the circle is   .

The circle is symmetrical about both the x-axis and y-axis . The radius of the circle is  .

Now,   

 

 

From Figure , the whole area enclosed by the given circle

Sq. units

Example 2 : Find the area enclosed by the ellipse

Solution:  The equation of the ellipse is

 The circle is symmetrical about both the x-axis and y-axis .

Now ,

    

   

  

 

From Given figure,

The area enclosed by the ellipse

Sq. units

Class 12 Maths Chapter 12 Application of integrals Exercise 8.1 Questions and Solutions

1. Find the area of the region bounded by the ellipse .

Solution : The equation of the ellipse is .

The ellipse is symmetrical about both the x-axis and y-axis .

Now,  

From figure ,

The area of the ellipse

Sq. units

2. Find the area of the region bounded by the ellipse .

Solution: The equation of the ellipse is .

The circle is symmetrical about both the x-axis and y-axis .

Now,  

 

From the given figure ,

The area of the ellipse

Sq. units

Choose the correct answer in the following exercise 3 and 4 .

3. Area lying in the first quadrant and bounded by the circle  and the lines and is

(A)      (B)       (C)           (D) 

Solution:  Given, the equation of circle is  and the lines and .

Now,  

Area of the first quadrant of the circle

Sq.units

The correct answer (A)  .

4. Area of the region bounded by the Curve -axis and the line  is

(A) 2     (B)           (C)         (D)  

Solution: Given, the curve is  , -axis and the line  .

Now,  

The area of the shaded region

The correct answer (B)  .

 Miscellaneous Exercise on Chapter  8

Example 3 Find the area of the region bounded by the line  , the x-axis and the ordinates  and   .

Solution: Given the line is  .

Given, x-axis ,i.e.,   , then 

In figure,

So, the graph lies below x-axis for the point and above x-axis for the point

Therefore, the area of the region 

Example 4 Find the area bounded by the curve  between  and  .

Solution: Given, the curve is 

From the given figure,

The required area = Area of the region OABO + Area of the region BCDB + Area of the region DEFD

Square units

Class 12 Maths Chapter 12 Application of integrals Miscellaneous Exercise Questions and Solutions

1. Find the area under the given curves and given lines :

(i)   and-axis

(ii)   and -axis

Solution: (i) Given, the curve is and the lines and-axis .

From given figure,

The area of the shaded region

Square units

(ii) Given, the curve is and the lines and -axis .

From given figure,

The area of the shaded region Square units.

2. Sketch the graph of  and evaluate 

Solution: Given, the curve is

In figure ,

If   , then

	– 4	– 5	– 6
	1	2	3

If   , then

	– 2	– 1	– 3
	1	2	0

Area of the shaded region

Square units.

3. Find the area bounded by the curve  between and  .

Solution: Given, the curve is.

In figure ,

Area of the shaded region = Area of OPQO + Area of QRSQ

 Square units

Choose the correct answer in the following Exercises from 4 to 5 .

4. Area bounded by the curve  , the-axis and the ordinatesand  is

(A)  – 9     (B)        (C)         (D)

Solution:  Given, the curve is

Area bounded by the curve 

Sq. units

The correct answer (D) .

5. The area bounded by the curve  ,-axis and the ordinates and is given by

(A)  0     (B)        (C)         (D)  

 [ Hint :  if  and  if   ]

Solution: Given, the curve is

Area of region sq. units

Correct Answer : (C)