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Subsections


MA 1024 Lab 5: Volumes and double integrals

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/Intdouble_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 Intdouble_start.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

\begin{displaymath}\int_R \! \int f(x,y) \, dA \end{displaymath}

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

\begin{displaymath}\int_a^b \left( \int_c^d f(x,y) \, dy \right) dx \end{displaymath}

or

\begin{displaymath}\int_c^d \left( \int_a^b f(x,y) \, dx \right) dy \end{displaymath}

where the rectangle $D$ 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.

Exercises

  1. Use Maple to compute the following double integrals.

    1. \begin{displaymath}\int_0^2 \int_2^5 x^2+3y \, dx \, dy\end{displaymath}


    2. \begin{displaymath}\int_{0}^{\pi} \int_{0}^{\pi} cos(xy)+2 \, dx \, dy\end{displaymath}


    3. \begin{displaymath}\int_{0}^{1} \int_{0}^{2} \frac{y}{1+x^2} \, dx \, dy\end{displaymath}


    4. \begin{displaymath}\int_{-3}^{-1} \int_{0}^{10} \exp(x+2y) \, dy \, dx\end{displaymath}


next up previous
Next: About this document ... Up: lab_template Previous: lab_template
William W. Farr
2001-12-07