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The area (in sq. units) of the region {(...

The area (in sq. units) of the region `{(x,y):y^(2)ge2x and x^(2)+y^(2)le4x,xle0,yge0}` is

A

`pi - (4)/(3)`

B

`pi - (8)/(3)`

C

`pi - (4sqrt(2))/(3)`

D

`(pi)/(2) - (2sqrt(2))/(3)`

Text Solution

Verified by Experts

The correct Answer is:
B
Given equations of curves are ` y^(2)=2x`
Which is a parabola with vertex (0,0) and axis parallel to X-axis. ` " ..(i)" `
And ` x^(2)+y^(2)=4x `
which is a circle with centre (2,0) and radius = 2 ` " …(ii) " `
On substituting ` y^(2)=2x ` in Eq. (ii), we get
` x^(2)+2x=4x rArr x^(2)=2x rArr x=0 " or " x=2 `
` rArr y=0 " or " y pm 2 " [using Eq.(i)]"`
Now, the required area is the area of shaded region, i.e.

` " Required area "=("Area of a circle")/(4) -int_(0)^(2)sqrt(2x)dx`
`=(pi(2)^(2))/(4)-sqrt(2) int_(0)^(2) x^(1//2)dx=pi-sqrt(2)[(x^(3//2))/(3//2)]_(0)^(2)`
`=pi - (2sqrt(2))/(3)[2sqrt(2)-0]=(pi-(8)/(3))` sq unit
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