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If the line x=alpha divides the area of ...

If the line `x=alpha` divides the area of region `R={(x,y) in R^(2): x^(3)leylex, 0le xle 1}` into two equal parts, then

A

`(pi)/(2)+(4)/(3)`

B

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

C

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

D

`(pi)/(2)+(2)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
D
`x^(2)+y^(2)le1" represents the rgeion inside the circle "x^(2)+y^(2)=1.`

`y^(2)le1-x`, represents the region inside parabola `y^(2)=1-x` towards focus.
The common region is as shown in the figure.
The area of shaded region
= Area of semicircle ADOBA+ Area of region ODCBO
`=(pi)/(2)+2overset(1)underset(0)int(1-y^(2))dy`
`=(pi)/(2)+2[y-(y^(3))/(3)]_(0)^(1)`
`=(pi)/(2)+2[1-(1)/(3)]`
`=(pi)/(2)+(2)/(3)`
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