Question

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.
y = x^3, y = 0, x = 1, x = 3

Answer 1

Using shells of thickness dx,
v = ∫[1,3] 2πrh dx
where r=x and h=x^3
v = ∫[1,3] 2πx^4 dx = 484π/5

Using discs of thickness dy, you have a cylindrical shell of height 1, with inner radius 1 and outer radius 3, plus the discs on up to y=27
v = π(3^2-1^2)*1 + ∫[1,27] π*(3^2 - ∛y^2) dy = 8π + 444π/5 = 484π/5