Area under y = -x^2 + 1 between x = 0 and x = 1 is rotated around x-axis. Find the volume by shell method. (Please solve it using step by step solution).
Each shell has thickness dy, so its volume is 2πrh dy. Add them all up and you have
v = ∫[0,1] 2πrh dy
where r=y and h=x=√(1-y)
v = ∫[0,1] 2πy√(1-y) dy = 8π/15
As a check, using discs,
v = ∫[0,1] πr^2 dx
where r=y=(1-x^2)
v = ∫[0,1] π(1-x^2)^2 dx = 8π/15