Question

How do you find the volume of a cone using an integral?

Answer 1

A cone with base radius `r` and height `h` can be obtained by rotating the region under the line `y=r/hx` about the x-axis from `x=0` to `x=h`.
By Disk Method,
`V=pi int_0^h(r/hx)^2 dx={pi r^2}/{h^2}int_0^hx^2 dx`
by Power Rule,
`={pir^2}/h^2[x^3/3]_0^h={pir^2}/{h^2}cdot h^3/3=1/3pir^2h`