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
\[R = \left\{ {\left( {x,y} \right)|\;a \le x \le b,\;\; p\left( x \right) \le y \le q\left( x \right)} \right\}.\]
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
\[R = \left\{ {\left( {x,y} \right)|\;u\left( y \right) \le x \le v\left( y \right),\;\; c \le y \le d} \right\}.\]
Figure 2.
Fubini's Theorem
Let \(f\left( {x,y} \right)\) is a continuous function over a type \(I\) region \(R\) such that
\[R = \left\{ {\left( {x,y} \right)|\;a \le x \le b,\;\; p\left( x \right) \le y \le q\left( x \right)} \right\}.\]
Then the double integral of \(f\left( {x,y} \right)\) in this region is expressed in terms of the iterated integral:
\[\iint\limits_R {f\left( {x,y} \right)dA} = \int\limits_a^b {\int\limits_{p\left( x \right)}^{q\left( x \right)} {f\left( {x,y} \right)dydx} } .\]
For a region of type \(II\) we have the similar theorem:
If \(f\left( {x,y} \right)\) is a continuous function on a type \(II\) region \(R\) such that
\[R = \left\{ {\left( {x,y} \right)|\;u\left( y \right) \le x \le v\left( y \right),\;\; c \le y \le d} \right\}.\]
then
\[\iint\limits_R {f\left( {x,y} \right)dA} = \int\limits_c^d {\int\limits_{u\left( y \right)}^{v\left( y \right)} {f\left( {x,y} \right)dxdy} } .\]
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 \[\int\limits_0^1 {\int\limits_1^2 {xydydx} }.\]
Solution.
We first evaluate the inner integral and then the outer integral:
