Question

Find the volume of the solid obtained by rotating the region bounded by the curves y=cos(X), y=0, x=0, x=pi/2 about the line y=-1

Volume=?

Answer 1

using discs of thickness dx,

v = ∫[0,π/2] πr^2 dx
where r = y = cosx
v = ∫[0,π/2] π(cosx)^2 dx = π^2/4

using shells of thickness dy,

v = ∫[0,1] 2πrh dy
where r=y and h = x = arccos(y)
v = ∫[0,1] 2πy*arccos(y) dy = π^2/4