Volumes of a Cone and a Cylinder

Date: 07/09/2003 at 07:19:00
From: Tony Francis
Subject: The volume of a cone and the volume of a cylinder

I am having trouble remembering formulae for areas/volumes. The volume 
of a cylinder seems obvious, the area of the circle * length. The 
volume of a cone appears to by one third of this. The fraction seems 
to come up in other areas: e.g. a circle and a sphere (four thirds). 

Can you tell me why the volume of a cone is a third of the volume of 
a cylinder? I suspect it is something related to calculus but I don't 
know where to start and the answer would probably help me remember the 
formulae.

Date: 07/09/2003 at 08:06:40
From: Doctor Jaffee
Subject: Re: The volume of a cone and the volume of a cylinder

Hi Tony,

Draw a segment whose endpoints are (r,0) and (0,h), where r,h > 0.  If 
you revolve this segment around the y-axis, it will form a cone whose 
base is a circle of radius r and whose height is h.

To calculate the volume, sketch a representative disk somewhere 
centered on the y-axis. The area of the surface of the disk shuld be 
pi*x^2, where the x-number is the radius of the circle. The thickness 
of the disk, then, is dy.

So, the volume of the cone will be the integral from 0 to h of 
pi*x^2 dy. Of course, you'll have to express x in terms of y to allow 
you to do the integration. The result should be V = pi*r^2*h/3.

Give it a try and if you want to check your solution with me or if you 
have difficulties or other questions, write back to me and I'll try to 
help you some more.

Good luck,

- Doctor Jaffee, The Math Forum
  http://mathforum.org/dr.math/