Question

Find the Volume V of the solid of revolution generated by revolving the region bounded by the x- axis and the graph of y=4x-x^2 about the line y=4

Answer 1

Using discs of thickness dx, we get
v = ∫[0,4] πr^2 dx
where r=y = 4x-x^2
v = ∫[0,4] π(4x-x^2)^2 dx = 512π/15

Using shells of thickness dy, and exploiting the symmetry of the region (as we could have done above, also)
v = 2∫[0,4] 2πrh dy
where r = y and h = 2-x = 2-(2-√(4-y)) = √(4-y)
v = 2∫[0,4] 2πy√(4-y) dy = 512π/15