Question

Find the volume of the solid generated by revolving region bound by y=x^2, y=0, and x=2 about the x axis.

Answer 1

Volume = ∫ y^2 dx from 0 to 2
= π∫ x^4 dx from 0 to 2

easy to integrate, and evaluate.

to check enter "π∫ x^4 dx from 0 to 2" into
https://www.wolframalpha.com/

Answer 2

using discs of thickness dx,
v = ∫[0,2] πr^2 dx = ∫[0,2] πx^4 dx = 32π/5
using shells of thickness dy,
v = ∫[0,4] 2πrh dy = ∫[0,4] 2πy(2-√y) dy = 32π/5