Find the area bounded by the curve `xy^(2)=4(2-x)` and y-axis.

Options

A. `2pi` sq units
B. `4pi` sq units
C. `12pi` sq units
D. `6pi` sq units

Answer

B

Explanation

In the equation of curve `xy^(2)=4(2-x)`, the degree of y is even. Therefore, the curve is symmetrical about X-axis and lies in `0ltxle2`.
The bounded area by the curve is `2int_(0)^(2)ydx`
`=2int_(0)^(2)2sqrt((2-x)/(x))dx=4int_(0)^(2)sqrt((2-x)/(x))dx`
Put `x=2sin^(2)thetaimpliesdx=4sintheta.costhetad theta`
`therefore A=4int_(0)^(pi//2)sqrt((2-2sin^(2)theta)/(2sin^(2)theta)).4sintheta.costhetad theta`
`=8[theta+sin.(2theta)/(2)]_(0)^((pi)/(2))=8[(pi)/(2)+0-0]=4pi` sq units