This preview shows pages 1–3. Sign up to view the full content.
Triple Integrals
Integration of a function of three variables, w=f(x,y,z), over a three
dimensional region R in xyzspace is called a triple integral and is denoted
Triple Integrals in BoxLike Regions
Suppose that R is the box with a<=x<=b, c<=y<=d, and r<=z<=s.
The triple integral is given by
To compute the iterated integral on the left, one integrates with respect to z
first, then y, then x. When one integrates with respect to one variable,
all other
variables are assumed to be constant
. For a boxlike region, the integral is
independent of the order of integration, assuming f(x,y,z) is continuous. Hence,
there are total of 6 ways to order the integrations. For example we can integrate
with respect to x, then z, then y. In this case we have
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Consider the following example:
The inner integral is
Integrating with respect to z, treating x and y as constants, we obtain
Note that z has completely disappeared from the expression on the right. The
middle integral is with respect to y and x is treated as a constant. We have
Note that y has disappeared from the expression on the right. The outer integral
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 01/17/2011 for the course MATH 1261 taught by Professor Staff during the Spring '08 term at GCSU.
 Spring '08
 Staff
 Calculus, Integrals

Click to edit the document details