Question

How do you find the volume bounded by y=1 and `y=x^2` revolved about the y=1?

Answer 1

`16/15 pi`

Explanation:

`R = 1 - x^2`

`R = 0 => x = ± 1`

`V = int_-1^1 pi R^2 text{ d}x`

`V = pi int_-1^1 (1 - x^2)^2 text{ d}x`

`V = pi int_-1^1 (x^4 - 2 x ^2 + 1) text{ d}x`

`V = pi [x^5/5 - 2 x ^3/3 + x]_-1^1`

`V = pi [1/5 - 2/3 + 1 - (-1/5 + 2/3 - 1)]`

`V = pi [2/5 - 4/3 + 2] = pi frac{6 - 20 + 30}{15}`