Question

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
y=x^2, y = 0, x = 0, x = 3, about the y-axis

Answer 1

use discs, stacked up around the y-axis, from y=0 to 9

V = Int(pi x^2 dy)[0,9]
= pi*Int(y dy)[0,9]
= pi * (1/2 y^2)[0,9]
= 81pi/2

You could also use shells of thickness dx:

V = Int(2pi * r * h dx)[0,3]
= 2pi Int(x * x^2 dx)[0,3]
= 2pi Int(x^3 dx)[0,3]
= 2pi x^4/4 [0,3]
= 2pi * 81/4
= 81pi/2

Answer 2

Actually, the height of the shells is (9-y), rather than y, but as it turns out, the final answer is the same!