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

Options

A. `(3pi)/(8)`
B. `(5pi)/(8)`
C. `(pi)/(2)`
D. `(pi)/(8)`

Answer

C

Explanation

Let A be the required area. Then,
A=4 (Area of the shaded region in first quadrant)
`implies A=4underset(0)overset(1//sqrt2)(int)(y_(1)-y_(2))dx`
`impliesA=4underset(0)overset(1//sqrt2)(int)(sqrt(1-x^(2))-x)dx`
`impliesA=4[(1)/(2)xxsqrt(1-x^(2))+1/2sin^(-1)x-(x^(2))/(2)]_(0)^(1//sqrt2)`

`impliesA=4[(1)/(2sqrt2)xx(1)/(sqrt2)+1/2xx(pi)/(4)-1/4]=pi` sq. units.