Iterated Integrals

    In this topic we evaluate many iterated integrals working through examples in a step-by-step manner. As with single integrals, it is often not practical to evaluate a double integral even over a simple rectangular region by using the definition. Instead, we compute double integrals by a process called iterated integration, which is like partial differentation in reverse. Suppose is continuous over the rectangle Then we write   to denote the integral obtained by integrating with respect to over the interval with held constant. The integral obtained by this partial integration is a function of alone,   which we integrate over the interval to obtain the iterated integral



Similarly, if we integrate first with respect to over holding constant, and then with respect to over we obtain the iterated integral,  



Fubini's Theorem says that these double integrals have the same value, provided is continuous over (in fact, Fubini's theorem say more).

Proposition (Fubini's Theorem) If is continuous over the rectangle then the double integral may be evaluated by either iterated integral; that is,  

Example (Iterated Integrals) Use iterated integration to compute the double integral   where

    Solution. We compute  

Example (Iterated Integrals) Use iterated integration to compute the double integral where

    Solution. We compute  

Example (Iterated Integrals) Use iterated integration to compute the double integral where

    Solution. We compute

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