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calc2 - Triple Integrals Integration of a function of three...

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Triple Integrals Integration of a function of three variables, w=f(x,y,z), over a three- dimensional region R in xyz-space is called a triple integral and is denoted Triple Integrals in Box-Like 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 box-like 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
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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
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