CALCULUS Understanding Its Concepts and Methods

Double and triple integrals, Fubini's theorem, Iterated integral

Suppose f(x,y)  is a continuous function on the rectangle R = [a,b] x [c,d]. If f(x,y) > 0 on the rectangle R, then the double integral of f over R is the volume of the solid above R and below the surface z = f(x,y).

If f(x,y) also takes on negative values, then the double integral is the difference between the volume where f is positive and the volume where f is negative.

The double integral of a function on a rectangle can be computed by an iterated integral:

Fubini's theoremIf is continuous on R = [a,b] x [c,d], thenThe integrals on the right above are iterated integrals.

We can write the triple integral over a rectangular solid E as the limit of a sum. To do this, subdivide E into small rectangular solids Si of volume Vi and let (xi , yi , zi) be any point in . Then the triple integral of f(x,y,z) over the rectangular solid E is the limit ofas the volumes Vi go to zero. The volume of E is the triple integral of f(x,y,z) = 1 over E.

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.