The following theorem gives a practical method for evaluating a double integral by expressing it as an iterated integral. 

 

Fubini’s Theorem

If  is continuous on the rectangle 

Then

More generally, this is true if we assume that  is bounded on ,  is discontinuous only on a finite number of smooth curves, and the iterated integrals exist.

 

 

EXAMPLE 8: Evaluate the double integral  where 

 

SOLUTION 1: Fubini’s Theorem gives

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SOLUTION 2: Again applying Fubini’s Theorem, but this time integrating with respect to x first, we have 

 

 

EXAMPLE 9: Evaluate the double integral , where 

 

 If we first integrate with respect to  , we get 

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The double integral of  can be written as the product of two single integrals: 

 

EXAMPLE 9: if , then