Calculus

Double Integrals

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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

R={(x,y)|axb,p(x)yq(x)}.
Region of integration of type I
Figure 1.

Definition 2.

A plane region R is said to be of type II if it lies between the graphs of two continuous functions of y (Figure 2), that is

R={(x,y)|u(y)xv(y),cyd}.
Region of integration of type II
Figure 2.

Fubini's Theorem

Let f(x,y) is a continuous function over a type I region R such that

R={(x,y)|axb,p(x)yq(x)}.

Then the double integral of f(x,y) in this region is expressed in terms of the iterated integral:

Rf(x,y)dA=abp(x)q(x)f(x,y)dydx.

For a region of type II we have the similar theorem:

If f(x,y) is a continuous function on a type II region R such that

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 Applying the Fubini's theorem we obtain

See more problems on Page 2.

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