Question

Use the disk method to find the volume of the solid generated by revolving about the y-axis the region bounded by the curves y=8−x^2 and the curve y=x^2.

Answer 1

Since the y-axis is also the axis of symmetry for the curves, we can just take one side and revolve it. This will be easiest using shells of thickness dx:

v = ∫[0,2] 2πrh dx
where r=x and h=(8-x^2)-x^2=8-2x^2
v = ∫[0,2] 2πx(8-2x^2) dx = 16π