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### Computing double integrals in Maple

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);
```

William W. Farr
Thu Apr 25 17:53:00 EDT 1996