Question

Find the volume?

Find the volume generated when the area bounded by y=sinx and the x-axis on 0<=x<=pi is rotated about the x-axis.

Answer 1

`pi^2/2`

Explanation:

`V = int_0^pi A(x) * dx`

`V = int_0^pi pi y^2 * dx`

`V = pi int_0^pi (sin x)^2 * dx`

`cos 2x = cos^2 x - sin^2 x`
`1 = cos^2 x + sin^2 x`

`1 - cos 2x = 2 sin^2 x`

`V = pi int_0^pi dx/2 - pi int_0^pi (cos 2x) /2 * dx`

`u = 2x => du = 2 * dx`

`V = pi/2 (pi - 0) - pi/2 int_0^{2 pi} cos u * (du)/2`

`V = pi^2/2 - pi/4 (0 - 0)`