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

The area (in sq. units) of the region described by `{(x,y),y^(2)le2x and yge4x-1}` is

A

`(7)/(32)`

B

`(5)/(64)`

C

`(15)/(64)`

D

`(9)/(32)`

Text Solution

Verified by Experts

The correct Answer is:
D
Given region is `{(x,y):y^(2)le 2x " and " yge 4x-1 } `
` y^(2)le 2x ` repressents a region inside the parabola
` y^(2)=2x " "…(i)`
and `y ge 4x-1` represents a region to the curves (i) and (ii) is
` (4x-1)^(2)=2x rArr 16x^(2)+1-8x=2x `
` rArr 16x^(2)-10x+11=0 rArr x=(1)/(2) ,(1)/(8)`
So, the points where these curves intersect are `((1)/(2),1) and ((1)/(8),(1)/(2)).`

`therefore " Required area "= int_(-1//2)^(1)((y+1)/(4)-(y^(2))/(2))dy `
` = (1)/(4)((y^(2))/(2)+y)_(-1//2)^(-1)-(1)/(6)(y^(3))_(-1//2)^(1) `
`=(1)/(4){((1)/(2)+1)-((1)/(8)-(1)/(2))}-(1)/(6){1+(1)/(8)}`
`=(1)/(4){(3)/(2)+(3)/(8)}-(1)/(6){(9)/(8)}`
`=(1)/(4)xx(15)/(8)-(3)/(16)=(9)/(32)` sq units
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