Question

Find the volume of the solid obtained by rotating the region bounded by y = 8 x 2 , x = 1 , and y = 0 , about the x -axis.

Answer 1

A cross-section is a disc with radius $8x^2$

$V = \int_0^1 A(x)\, dx = \int_0^1 \pi\left(8x^2\right)^2 dx = \int_0^1 64 \pi x^4 = \left[\frac{64\pi}{5} x^5\right]_0^1 \\ \\ = \frac{64\pi}{5}$