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

1 Answer
Wataru
Sep 8, 2014

A cone with base radius rr and height hh can be obtained by rotating the region under the line y=r/hxy=rhx about the x-axis from x=0x=0 to x=hx=h.
By Disk Method,
V=pi int_0^h(r/hx)^2 dx={pi r^2}/{h^2}int_0^hx^2 dxV=π∫h0(rhx)2dx=πr2h2∫h0x2dx
by Power Rule,
={pir^2}/h^2[x^3/3]_0^h={pir^2}/{h^2}cdot h^3/3=1/3pir^2h=πr2h2[x33]h0=πr2h2⋅h33=13πr2h

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