Subsections

• Purpose
• Getting Started
• Background
  • Volumes from double integrals
  
  
• Exercises

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 /math/calclab/MA1024/Doubleint_start_D09.mws ~/My_Documents

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_D09.mws, and go through it carefully. Then you can start working on the exercises.

Background

Volumes from double integrals

Suppose that $R$ is a rectangular region in the the $x-y$ plane, and that $f(x,y)$ is a continuous, non-negative function on $R$. Then the volume of the solid above $R$ and below $z=f(x,y)$ is given by the double integral

$$\int_R \! \int f(x,y) \, dA$$

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

$$\int_a^b \left( \int_c^d f(x,y) \, dy \right) dx$$

or

$$\int_c^d \left( \int_a^b f(x,y) \, dx \right) dy$$

where the rectangle $R$ is defined by the inequalities $a \leq x \leq b$ and $c \leq y \leq d$.

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 $R = \left\{(x,y): g_1(x) \leq y \leq g_2(x), a \leq x \leq b \right\}$. This is the case where the base of the solid is not rectangular, but is bounded by two curves $y=g_1(x)$ and $y=g_2(x)$. If $f(x,y)$ is as before, then the volume of the solid above $S$ and below $z=f(x,y)$ is given by

$$\int_a^b \left( \int_{g_1(x)}^{g_2(x)} f(x,y) \, dy \right) dx$$

The other case, where the region $R$ is $x$-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.

   a)

   
   

   $$\int_2^5 \int_{-1}^4 x^2(y^2-4y) \, dx \, dy$$

   
   

   b)

   
   

   $$\int_{-1}^{2} \int_{-2}^{1} \frac{y+1}{x^2+1} \, dy \, dx$$

   
   

   c)

   
   

   $$\int_{-4}^{-2} \int_{0}^{4} exp(2y-3x) \, dx \, dy$$

   
   

2. Let $R$ be the region in the $xy$ plane bounded by the two curves $y=\sqrt{9-x^2}$ and $$y= -\frac{1}{3}x+3$$. Use a double integral to compute the area of the region. Use both orders of integration. Include a plot of the region $R$ in your worksheet.
3. Use a double integral to find the volume of the region bounded by the two paraboloids $z=x^2+3y^2$ and $z=16-2x^2-y^2$.
4. Use a double integral to prove that the volume of a sphere of radius $r$ is $$\frac{4}{3}\pi r^3$$.