Iterated Integrals
Regions of Type I and Type II
The most powerful tool for the evaluation of the double integrals is the Fubini's theorem. It works not for a general region R but for some special regions which we call Regions of type I or type II.
Definition 1.
A plane region R is said to be of type I if it lies between the graphs of two continuous functions of x (Figure 1), that is
Definition 2.
A plane region
Fubini's Theorem
Let
Then the double integral of
For a region of type
If
then
Thus, the Fubini's theorem allows to calculate double integrals through the iterated ones. To evaluate an iterated integral, we first find the inner integral and then the outer integral.
Solved Problems
Example 1.
Evaluate the iterated integral
Solution.
We first evaluate the inner integral and then the outer integral:
Example 2.
Find the iterated integral
Solution.
Here we have the region of type