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QUESTION: 1

The area lying in the first quadrant and bounded by the curve y = x^{3} , the x – axis and the ordinates at x = - 2 and x = 1 is

Solution:

Required area :

QUESTION: 2

Area bounded by the curves satisfying the conditions is given by

Solution:

QUESTION: 3

The area of the figure bounded by y = e^{x}, y = e^{−x} and the straight line x = 1 is

Solution:

QUESTION: 4

The area bounded by the parabolas y = (x+1)^{2} and y = (x−1)^{2} and the line y = (1/4) is equal to

Solution:

Required area :

QUESTION: 5

The area enclosed between the curve y = log_{e}(x+e) and x = log_{e} 1/y and the x- axes is

Solution:

Required area is :

QUESTION: 6

The area bounded by the curve y = x^{3}, the x – axis and two ordinates x = 1 and x = 2 is

Solution:

QUESTION: 7

The area bounded by the parabola y = x^{2} + 1 and the straight line x + y = 3 is given by

Solution:

The two curves parabola and the line meet where,

Required area ;

QUESTION: 8

The area enclosed between the curves y = √x , x = 2y+3and the x-axis is

Solution:

The two curves meet where;

Therefore, the two curves meet where x = 9.

Therefore,required Area:

QUESTION: 9

The area of the region {(x , y) : x^{2}+y^{2}⩽1⩽x+y} is equal to

Solution:

x^{2}+y^{2} = 1,x+y = 1

Meets when

x^{2}(1−x)^{2} = 1

⇒ x^{2}+1+x^{2}−2x = 1

⇒ 2x^{2}−2x= 0 ⇒ 2x(x−1)=0

⇒ x = 0,x = 1.

i.e. points (1 ,0) ,(0 ,1). Therefore , required area is ;

QUESTION: 10

The area bounded by y = |sinx| , the x – axis and the line |x| = π is

Solution:

Required area :

QUESTION: 11

If A is the area between the curve y = sin2x , x – axis and the lines x =

Solution:

Required area :

QUESTION: 12

The area bounded by the curves y^{2} = x and y = x^{2} is

Solution:

The two curves meet in (0 , 0) and (1, 1).The required area lies above the curve y = x^{2} and below x = y^{2} and is equal to ;

QUESTION: 13

The positive value of the parameter a for which the area of the figure bounded by y = sin ax , y = 0 , x = x/a and x = x/3a is 3 is equal to

Solution:

We have :

QUESTION: 14

The area bounded by the parabolas y = 5x^{2}and y − 9 = 2x^{2} is

Solution:

Required area :

QUESTION: 15

The area bounded by the curves y = √x , 2y+3 = x and the x – axis in the first quadrant is

Solution:

Required area :

QUESTION: 16

The area bounded by y = 2cosx , x = 0 to x = 2π and the axis of x in square units is

Solution:

Required area :

Therefore , total area from x = 0 to x = 2π is 4 X 2= 8 sq. units.

QUESTION: 17

The area common to the circle x^{2}+y^{2} = 16a^{2} and the parabola y^{2} = 6ax is

Solution:

Required area :

QUESTION: 18

The area bounded by the parabola y^{2} = 4x and the line x + y = 3 is

Solution:

Required area :

QUESTION: 19

The area enclosed by the parabola y^{2 }= 2x and its tangents through the point (-2 , 0) is

Solution:

QUESTION: 20

The area bounded by the curve y = x log x and y = 2x−2x^{2} is

Solution:

Required area :

QUESTION: 21

The area of the figure bounded by the curve y = log_{e}x , the x – axis and the straight line x = e is

Solution:

Required area :

QUESTION: 22

The area of the region bounded by the curves y = |x−1| and y = 3 - |x| is

Solution:

Required area :

QUESTION: 23

The area bounded by the curve y = x^{2}+1and the line x + y = 3 is

Solution:

The two curves parabola and the line meet where,

Therefore , the required area is :

QUESTION: 24

The area bounded by the angle bisectors of the lines x^{2}−y^{2}+2y = 1 and x+y = 3 is

Solution:

The angle bisectors of the line given by x^{2}−− y^{2}+2y = 1 are x = 0 , y = 1. Required area = 1/2 .2.2 = 2 sq. units

QUESTION: 25

The area bounded by the curves y= |x−1| and y = 1 is given by

Solution:

The given curves are : (i) y = x – 1 , x > 1 . (ii) y = - (x – 1) , x < 1. (iii) y = 1 these three lines enclose a triangle whose area is : 1/2 .base.height = 1/2 .2 .1 = 1 sq. unit.

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