Subsections

• Purpose
• Getting Started
• Background
  • Computing double integrals
  
  
• Exercises

MA 1024 Lab 5: Double integrals

Purpose

The purpose of this lab is to acquaint you with using Maple to evaluate 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

Computing double integrals

Suppose that $R$ is a rectangular region in the the $x-y$ plane, and that $f(x,y)$ is a continuous function on $R$. Then the double integral of $f$ over $R$ is denoted by

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

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 over rectangular regions.

It is also possible to use Maple to compute double integrals over regions that are not rectangles. The only hard part is setting up the limits of integration. The worksheet contains examples of how to use Maple to help you do this.

Exercises

1. Use Maple to compute the following double integrals.
   1. 
      
      $$\int_{0}^{\pi} \int_{0}^{\pi} \cos(xy)+2 \, dx \, dy$$
      
      

   2. 
      
      $$\int_{0}^{1} \int_{0}^{2} \frac{y}{1+x^2} \, dx \, dy$$
      
      

   3. 
      
      $$\int_{-3}^{-1} \int_{0}^{1} \exp(x+2y) \, dy \, dx$$
      
      

2. Consider the following integral
   
   $$\int_S \! \int x^2+y^3 \, dA$$
   
   
   where the region $S$ is bounded by $2x+3y=6$, $x=-2$, and $y=0$. Compute the integral using $x$ as the inner variable of integration and then repeat the calculuation using $y$ as the inner variable of integration. You should get the same answer.