Consider the solid obtained by rotating the region bounded by the given curves about the x-axis.

Question

y = 9 - 9x^2 , y = 0
Find the volume V of this solid.

Sketch the region, the solid, and a typical disk or washer.

Any help or tips would be greatly appreciated; not sure how to work it :/

Answer 1

The curve intersects the x-axis at (1,0) and (-1,0).

Volume = π[integral] y^2 dx from -1 to 1
or by symmetry
= 2π[integral] (9-9x^2)^2 dx from 0 to 1
= 2π[integral] (81 - 162x^2 + 81x^4) dx from 0 to 1
= 2π(81x - 54x^3 + (81/5)x^5) from 0 to 1
= 2π(81 - 54 + 81/5 - 0)
= 432π/5