Math Calc 1 Last question
posted by Anonymous .
Find the volume of the solid obtained by rotating the region bounded by the curves y=4xx^2, y=8x2x^2 about the line x=2.
I got 256 pi/3 but pretty sure my work is wrong.

straightforward shell problem  each shell has radius R = x + 2
and height h = 8x  2x²  (4x  x²) = 4x  x²,
and the range is 0 ≤ x ≤ 4, so
V = ∫[a,b] 2pi Rh dx = 2pi ∫[0,4] (x + 2)(4x  x²) dx = 2pi ∫[0,4] (2x²  x^3 + 8x) dx
V = 2pi ((2/3)x^3  (1/4)x^4 + 4x²) [0,4] = 2pi (128/3  64 + 64) = 256pi/3
barring computational error.
7 months ago