Suppose that **D** is a rectangular region in the the **x-y** plane, and
that is a continuous, non-negative function on **D**. Then the
volume of the solid above **D** and below is given by the
double integral

We learned in class that such integrals can be evaluated by either of the iterated integrals

or

where the rectangle **D** is defined by the inequalities and .

Using Maple to evaluate double integrals proceeds in exactly the same fashion. For example, the following Maple commands show the computation of

for both orders of integration.

> int(int(x^2+y^2,x=-2..1),y=0..1);

> int(int(x^2+y^2,y=0..1),x=-2..1);

More complicated integrals are handled similarly. Suppose that we have
and we want to know the volume of the solid
between the region in the **xy** plane bounded by the circle
and the graph of . Then an appropriate
integral is

Using Maple, we calculate this as

> int(int(x^2+y^2,y=-sqrt(1-(x-1)^2)..sqrt(1-(x-1)^2)),x=0..2);

Thu Apr 25 17:53:00 EDT 1996