Find the volume formed by rotating about the y-axis the region enclosed by x=11y and y^3=x with y≥0

Answer 1

The curves meet at (0,0) and (11√11,√11)
So, using shells of thickness dx,
v = ∫[0,11√11] 2πrh dx
where r=x and h = ∛x - x/11
v = ∫[0,11√11] 2πx(∛x - x/11) dx = 5324/21 π√11

using washers of thickness dy,
v = ∫[0,√11] π(R^2-r^2) dy
where R = 11y and r = y^3
v = ∫[0,√11] π((11y)^2-y^6) dy = 5324/21 π√11