The region in the first quadrant bounded by the graph of y

The region in the first quadrant bounded by the graph of y = sin x2 , the coordinate axes, and the line x = b is revolved about the y-axis. Find b such that the volume of the solid generated is pi/2 cubic inches.

Solution

V = integral of 2rf(x)dx.......limit from 0 to b
r = x, f(x) sinx^2
V = integral of 2xsinx^2dx
let x^2 = t
2xdx =dt
xdx = dt/2
V = 1/2 integral of 2sintdt
/2 = 1/2 integral of 2sintdt
integrating right hand side
1 = -2cost
apply limits
1 = -2(cosb - 1)
1 = -2cosb + 2
-1 = -2cosb
cosb = 1/2
b = /3