The base of a solid is the region bounded by the parabola x^2 = 8y and y=4. Each cross section perpendicular to the y-axis is an equilateral triangle. Find the volume.

Answer 1

Let's use a little symmetry here.
So, we have the base with length 2x and height x√3. That means if we add up all the thin slices, the volume is

v = ∫[0,8] (1/2)(2x)(x√3) dy
= ∫[0,8] √3 x^2 dy

But x^2 = 8y, so

v = ∫[0,8] 8√3 y dy

now just crank it out