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16.2 Iterated Integrals

Recall that it is usually difficult to evaluate single integrals directly from the definition of an integral, but the Fundamental Theorem of Calculus provides a much easier method. The evaluation of double integrals from first principles is even more difficult, but in this section we see how to express a double integral as an iterated integral, which can then be evaluated by calculating two single integrals.

Suppose that is a function of two variables that is continuous on the rectangle . We use the notation to mean that x is held fixed and is integrated with respect to y from to Hence we integrate y while treating x as a constant, this is called partial integration with respect to y.

Now we integrate the function A with respect to x from to , we get:

Above, the integral on the right is called an iterated integral. Usually the brackets are omitted. Thus:

In a similar manner:

These iterated integrals mean that we first integrate with respect to one variable (while holding the other fixed) and then integrating with respect to the other variable while holding the first one fixed.

The following theorem gives a practical method for evaluating a double integral by expressing it as an iterated integral (in either order):

Fubini's Theorem

If is a 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 integral exist.

In the special case where and Then Fubini's Theorem gives

In the inner integral is a constant, so is a constant and we can write

 

since is a constant.

Therefore, in this case, the double integral of can be written as the product of two single integrals:

 

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