Volumes of a Cone and a CylinderDate: 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/ |
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