The arc of the parabola y=x^2 from (1,1) to (2,4) is rotated about the y-axis. Find the area of the resulting surface.

Answer 1

using discs,

v = ∫[1,4] πr^2 dy
where r = x, so r^2 = y

v = π∫[1,4] y dy
= π (1/2 y^2)[1,4]
= π(8 - 1/2) = 15π/2

using shells, we need to add in the cylinder of radius 1 and height 3, volume 3π, which lies inside the curve

v = 3π + ∫[1,2] 2πrh dx
where r = x, h = 4-y = 4-x^2
v = 3π + 2π∫[1,2] x(4-x^2) dx
= 3π + 2π(2x^2 - 1/4 x^4)[1,2]
= 3π + 2π[(8-4)-(2-1/4)]
= 3π + 2π(4 - 7/4)
= 15π/2