Question

Determine the volume of the solid obtained by rotating the region bounded by y= x - x^3, x = 0, x = 1 and the x - axis about the y-axis.

Answer 1

Using shells (cylinders) of thickness dx, we have

v = ∫[0,1] 2πrh dx
where r = x and h = y
v = ∫[0,1] 2πx(x-x^3) dx = 4π/15