The area enclosed between the curve `y^(2)(2a-x)=x^(3)` and the line x=2 above the x-axis is

Options

A. `pia^(2)` sq. units
B. `(3pia^(2))/(2)` sq. units
C. `2pia^(2)` sq. units
D. `3pia^(2)` sq. units

Answer

B

Explanation

The required area `A=int_(0)^(2a)sqrt((x^(3))/(2a-x))dx`

`"Put "x=2a sin^(2) theta`
`rArr" "dx=2a sin theta cos theta d""theta`
`rArr" "A=8a^(2)int_(0)^(pi)((1-cos 2 theta)/(2))^(2)d""theta`
`=2a^(2)int_(0)^(pi)(1-2cos 2 theta + cos^(2)2theta)d""theta`
`2a^(2)int_(0)^(pi)(1-2 cos 2theta+(1+cos 4theta)/(2))d""theta=(3pia^(2))/(2)`