Question

Use method of shells to find the volume of a cone with raidius r and height h

Answer 1

hy..

A(x)A(x) is the area of a moving cross-section obtained by slicing through xx perpendicular to the xx-axis.

So a cross section of a cone is a circle, with area A(r)=πr2A(r)=πr2. But I know that computing the integral

V=∫0hπr2 .

, which is 1/3πr2h.

hope it helps

Answer 2

A cone with base radius r and height h can be obtained by rotating the region under the line.

$\tt y = \frac{rx}{h}$

About the x-axis from x=0 to x=h.

By Disk method.

$\tt v = \pi( \frac{r}{h} x)^{2} dx = \frac{\pi {r}^{2} }{ {h}^{2} } {x}^{2}dx$

By power rule.

$\tt = \frac{\pi {r}^{2} }{ {h}^{2} }( \frac{ {x}^{3} }{3} )^{h} = \frac{\pi {r}^{2} }{ {h}^{2} } . \frac{ {h}^{3} }{3} = \frac{1}{3} \pi {r}^{2} h$