Question

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.
xy = 2, x = 0, y = 2, y = 4

Answer 1

Each shell has thickness dy and height x = 2/y

So, the volume v is

v = ∫[2,4] 2πrh = ∫[2,4] 2πy(2/y) dy
= ∫[2,4] 4π dy
= 8π

As a check, you can use discs, getting

v = π(4^2-2^2)(1/2) + ∫[1/2,1] π(R^2-r^2) dx
where R=y and r=2
v = 6π + ∫[1/2,1] π((2/x)^2-2^2) dx
v = 6π+2π = 8π