Question

y = 32 − x^2, y = x^2; about x = 4
Trying to find V by using the method of cylindrical shells.
I got 1280pi/3, but my homework says it's wrong.
I did V=∫[0,4] 2πrh dx, which got me to V=∫[0,4] 2π(4-x)(32-2x^2)dx, hence 1280pi/3. But it's wrong and I'm stuck on it.

Answer 1

You need to integrate from -4 to 4. Do that and you will get 4096π/3

let's check with discs of thickness dy. The boundary changes at (4,16)
v1 = ∫[0,16] π((4+√y)^2 - (4-√y)^2) dy = 2048π/3
v2 = ∫[16,32] π((4+√(32-y))^2 - (4-√(32-y))^2) dy = 2048π/3