Having recently learned the disk/shell/washer method for finding the volume of a solid of revolution, I'm trying to apply similar methods to derive the formula for the surface area of a cone (and hopefully after that, that of a sphere).
The region that is revolved around to form the cone is that under f(x) = (r/h)x from 0 to h, where r is the radius of the base and h is the height of the cone (both are constants).
Since V = πr^2h for a cylinder, the volume of the cone is ∫ π[f(x)]^2 dx from 0 to h. When I evaluate that, I get V = πr^2h/3,which is correct.
The Attempt at a Solution
I reasoned that since A = 2πrh for the curved surface of a cylinder, evaluating ∫ 2πf(x) dx from 0 to h should result in an expression for the area of the curved surface of the cone (everything but the base). But instead of getting πrs (s being √(r^2 + h^2), I think it's called the lateral height), I get πrh as the area of the curved surface.
Is my method wrong, or am I just integrating wrong?