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Subsections


Cylindrical and Spherical Coordinates

Getting Started

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

cp /math/calclab/MA1024/Coords_start.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 Coords_start.mws, and go through it carefully. Then you can start working on the exercises.

Background

Defining surfaces with rectangular coordinates often times becomes more complicated than necessary. A change in coordinates can simplify things. Cylindrical coordinates can simplify plotting a region in space that is symmetric with respect to the $z$-axis such as paraboloids and cylinders. The paraboloid $z=x^2+y^2$ would become $z=r^2$ and the cylinder $x^2+y^2=1$ would become $r=1$. Spherical coordinates would simplify the equation of a sphere, such as $x^2+y^2+z^2=1$, to $\rho=1$. The conversion tables below show how to make the change of coordinates.

To change to cylindrical coordinates from rectangular coordinates use the conversion:

$x=r\cos(\theta)$
$y=r\sin(\theta)$
$z=z$

Where $r$ is the radius in the x-y plane and $\theta$ is the angle in the x-y plane.

To change to spherical coordinates from rectangular coordinates use the conversion:

$x=\rho\sin(\phi)\cos(\theta)$
$y=\rho\sin(\phi)\sin(\theta)$
$z=\rho\cos(\phi)$

Where $\theta$ is the angle in the x-y plane; $\rho$ is the radius from the origin in any direction; and $\phi$ is the angle in the x-z plane.

Using a triple integral to find the volume of a solid translates in the following manner:

\begin{displaymath}\int \int \int 1 \,dz \,dy \,dx = \int \int \int r \,dz \,dr ...
...ta = \int \int \int \rho^2\sin(\phi) \,d\rho \,d\phi \,d\theta \end{displaymath}

Exercises

  1. Given the rectangular equation for a circular cone:

    \begin{displaymath}
z=\sqrt{x^2+y^2}
\end{displaymath}

    A)
    Graph the equation using the domain values of $-10 \leq x \leq 10$, $-10 \leq y \leq 10$ and the range values $0 \leq z \leq 10$.
    B)
    Write the equation in cylindrical coordinates and then graph the equation.
    C)
    Write the equation in spherical coordinates and graph it.
    D)
    Use the simplest of the three coordinate systems and a triple integral to find the volume of a right circular cone of height $10$.

  2. Now suppose an icecream cone is bounded below by the same equation of the cone given in exercise 1 and bounded above by the sphere $x^2+y^2+z^2=100$. Plot the icecream cone using rectangular coordinates over the same intervals as exercise 1a. Find the volume of the icecream cone using spherical coordinates.

next up previous
Next: About this document ... Up: lab_template Previous: lab_template
Dina Solitro
2007-04-18