8

8

Q1. Find the area of the region bounded by the curves x² + y² = 2 and x = y².

Solution

The area above x axis is bounded between X-axis, curve C₂ from (0,1) and curve C₁from(1,

square root of 2

) and the total bounded area is symmetric about X-axis so the required area is

2 asterisk times left curly bracket integral subscript 0 superscript 1 C 2 d x plus integral subscript 1 superscript square root of 2 end superscript C 1 d x right curly bracket

Q2. Draw a sketch of the curves y = sin x and y = cos x at

and find the area of the region enclosed by them and X-axis.

Solution

The region is divided into two parts, one from x = 0 to and second from to x = .But the region is symmetric about x=∏/4 so the required area is

2 asterisk times integral subscript 0 superscript pi divided by 4 end superscript C 1 d x

2 asterisk times integral subscript 0 superscript pi divided by 4 end superscript sin left parenthesis x right parenthesis d x 2 asterisk times left square bracket negative cos left parenthesis x right parenthesis right square bracket subscript 0 superscript pi divided by 4 end superscript 2 asterisk times left square bracket 1 minus 1 divided by square root of 2 right square bracket 2 minus square root of 2

The required area is 2-

square root of 2 s q space u n i t

Q3. Find the area between the curves

.

Solution

T h e space r e q u i r e d space a r e a space i s equals integral subscript 0 superscript 8 left parenthesis C 1 minus C 2 right parenthesis d x integral subscript 0 superscript 8 left parenthesis 2 square root of 2 x minus x squared divided by 8 right parenthesis d x left square bracket square root of 2 x squared minus x cubed divided by 24 right square bracket subscript 0 superscript 8 64 square root of 2 minus 64 divided by 3 64 over 3 open parentheses 3 square root of 2 minus 1 close parentheses s q space u n i t

Q4. Find the area of the circle

exterior to the parabola

.

Solution

Q5. Find the area bounded by the curve

and X-axis.

Solution

The region bounded by two parallel lines x = 1 and x = 3.

Required area =

Q6. Find the area of region bounded by

, x = 1, x = 4 and the X-axis.

Solution

Q7. Find the area of region bounded by

and

.

Solution

Q8. Using integration, find the area bounded by the curve

.

Solution

Q9. Find the area of region bounded by

and

using integration.

Solution

Q10. Find the area enclosed by the curve

, x = 6, x = 9 and X-axis.

Solution

The curve, x = 6, x = 9.

Required area =.

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