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The area enclosed by the curve y=sqrt(4-...

The area enclosed by the curve `y=sqrt(4-x^2),ygeqsqrt(2)sin((xpi)/(2sqrt(2)))` , and the x-axis is divided by the y-axis in the ratio.
(a) `(pi^2-8)/(pi^2+8)`
(b) `(pi^2-4)/(pi^2+4)`
(c)`(pi-4)/(pi-4)`
(d) `(2pi^2)/(2pi+pi^2-8)`

A

`(pi^(2)-8)/(pi^(2)+8)`

B

`(pi^(2)-4)/(pi^(2)+4)`

C

`(pi-4)/(pi-4)`

D

`(2pi^(2))/(2pi+pi^(2)-8)`

Text Solution

Verified by Experts

The correct Answer is:
D
`y=sqrt(4-x^(2)),y=sqrt(2)sin ((xpi)/(2sqrt(2)))" intersect at "x=sqrt(2)`

Area of the left of y-axis is `pi`
Area to the right of y-axis
`=overset(sqrt(2))underset(0)int(sqrt(4-x^(2))-sqrt(2)sin""(xpi)/(2sqrt(2)))dx`
`=((xsqrt(4-x^(2)))/(2)+(4)/(2)sin^(-1)""(x)/(2))_(0)^(sqrt(2))+((4)/(pi)cos""(xpi)/(2sqrt(2)))_(0)^(sqrt(2))`
`=(1+2xx(pi)/(4))+(4)/(pi)(0-1)`
`=1+(pi)/(2)-(4)/(pi)`
`=(2pi-pi^(2)-8)/(2pi)` sq. units
`therefore" Ratio"=(2pi^(2))/(2pi+pi^(2)-8)`.
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