Subsections

# Double integrals with Maple

## Purpose

The purpose of this lab is to acquaint you with using Maple to do double integrals.

## Getting Started

To assist you, there is a worksheet associated with this lab that contains examples. You can copy that worksheet to your home directory with the following command, which must be run in a terminal window, not in Maple.

cp ~bfarr/Doubleint_start.mws ~


You can copy the worksheet now, but you should read through the lab before you load it into Maple. Once you have read to the exercises, start up Maple, load the worksheet Doubleint_start.mws, and go through it carefully. Then you can start working on the exercises.

## Background

### Volumes from double integrals

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

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

or

where the rectangle is defined by the inequalities and .

The worksheet associated with this lab contains examples of how to use Maple to compute double integrals. It also has an example of how to use Maple if the region of integration is of the more complicated form . This is the case where the base of the solid is not rectangular, but is bounded by two curves and . If is as before, then the volume of the solid above and below is given by

The other case, where the region is -simple can also be handled using Maple, and there is an example in the worksheet.

## Exercises

1. Use Maple to compute the following double integrals. In the second integral, can you explain why the result is zero even though is a positive function?

2. Consider the following integral

where the region is bounded by , , and . Compute the integral using as the inner variable of integration and then repeat the calculuation using as the inner variable of integration. You should get the same answer. Include a plot of the region in the plane.

3. Let be the region in the plane containing the origin and bounded by the two curves and . Compute

where .

4. Use a double integral to find the volume of the region beneath the surface and above the region in the plane . Include a plot of the region in the plane.