Question

Find the volume V of the solid obtained by rotating the region bounded
by curves y=x and y= √𝒙 about the x-axis.

Answer 1

the curves intersect at (0,0) and (1,1)
using shells of thickness dy,
v = ∫[0,1] 2πrh dy
where r = y and h = y - y^2
v = ∫[0,1] 2πy(y-y^2) dy = π/6

using discs (washers) of thickness dx,
v = ∫[0,1] π(R^2-r^2) dx
where R = √x and r=x
v = ∫[0,1] π(x-x^2) dx = π/6