CALCULUS Understanding Its Concepts and Methods

Volume by cross sections, Disc method, Shell method, Washer method

A solid of revolution is a solid figure obtained by rotating a plane region about a straight line that lies in the same plane. The term cross section is used to describe a slice of a solid of revolution, the slice being perpendicular to the axis of revolution. The volume of the solid of revolution can be computed by summing (possibly with an integral) the volumes of all of its slices.

If the region bounded by the graph of a nonnegative, continuous function
*f*,
the
*x*-axis,
and the interval
[*a*, *
b*],
is rotated about the
*x*-axis,
then each slice has radius
*f*(*x*)
and volume
p(*f*(*x*))^{2}*dx*.
The volume is given by the integral

This is sometimes called the disc method for computing volumes.

If a curve given by a function
*x* = *f*(*y*)
is revolved about the
*y*-axis
for
,
replace
*x*
by
*y*
in the expression above to find the volume of the solid of revolution.

If the region bounded by two functions
and an interval
[*a*, *
b*]
is rotated about the
*x*-axis,
then each slice has volume the difference
and the volume of the solid of revolution is given by the
integralThis
is sometimes called the washer method for
computing volumes.

If the region bounded by the graph of a nonnegative, continuous function
*f*,
the
*x*-axis,
and the interval
[*a*, *
b*]
with
,
is rotated about the
*y*-axis,
then each shell has radius
*x*
and volume
2p*xf*(*x*)
*dx*.
The volume is given by the integral

This is sometimes called the shell method for computing volumes.

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Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.