Math Shorts - Integrals by Metin Bektas

Author:Metin Bektas [Bektas, Metin]
Language: eng
Format: epub
Published: 2014-08-03T00:00:00+00:00

Volume - Basics

Integrals can also be used to calculate the volume of solids of revolution. These are solids that are obtained by rotating a plane curve y = f(x) around the x-axis. In the image below you can see the solid of revolution that results from the rotation of a straight line.

To derive a formula, it helps to think in infinitesimal sections. We divide the solid of revolution into very small cylinders of length dx and radius r. The radius is obviously given by the local value of the function, so r = f(x). According to the volume formula for cylinders, the volume of such an infinitesimal cylinder is dV = π·r2·dx = π·f(x)2·dx. Integrating both sides leads to this neat formula for the volume of the solid of revolution between the limits x = a and x = b:

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