A nice example of setting up integrals in cylindrical coordinates:

Find the mass of a 2 meter diameter, 4 meter long cylinder with
rounded out ends, if the ends are rounded out in the shape of spheres
of diameter 2 meters and the density of the cylinder is equal to 5
minus the distance along its length (in
kg/m^{3}).

Let's think about the cylinder on end, as shown in figure 1 to the right. This is reproduced in figure 2 below that, with the sides of the cylinder rendered only as wireframes so that the spherical missing parts at either end are easier to see.

How can we describe this region? The outside of the cylinder is obviously given by a circle of radius one, or

The top and bottom of the region are given by spheres of radius one
centered at the origin (0,0,0) and a point 4 meters up the *z*
axis, (0,0,4). But this isn't that hard -- we know the equation
of a sphere of radius *r* centered at the point
(*x*_{0},*y*_{0},*z*_{0}) is just

(*x* - *x*_{0})^{2} +
(*y* - *y*_{0})^{2} +
(*z* - *z*_{0})^{2} =
*r*^{2},

So the spheres at the top and bottom of the cylinder are just

respectively. Solving for *z* in either case, and noticing that
the for the top sphere we need the *bottom* half of the sphere,
and so take the negative sign when taking the square root, we get

Now, setting up the integral is straightforward. Let's pick our outer
variables in the triple integral to be *r* and *theta*.
To determine the limits on these variables, we look at the projection
of the figure into the *xy*-plane, which is just a circle of
radius one. So the limits on *r* and *theta* are just
0 <= *r* <= 1 and
0 <= *theta* <= 2 pi.

Then to determine the limits for the innermost integral, *z*, we
look at each point (*r*, *theta*) and determine what range of
values for *z* we are interested in. In this case, for any
(*r*, *theta*) the smallest value of *z* that we want to
consider is one that is on the bottom sphere. Then we want to
consider all values of *z* between this and the top sphere. So
the limits on *z* are

sqrt(1 - (*x*^{2} +
*y*^{2})) <= *z* <=
4 - sqrt(1 - (*x*^{2} +
*y*^{2})),

where *x*^{2} +
*y*^{2} is of course equal to
*r*^{2}. Then the full integral
is, filling in the density *d* = 5 - *z*,

Evaluating this, we get 8 pi (in kg).

Why did we use cylindrical coordinates here? (Other than because the
title was "cylindrical coordinates", of course.) The region in the
*xy*-plane over which we're integrating is nicely defined with
polar coordinates, and the limits on the domain in the remaining
(*z*) direction are nice (at least, relatively nice) functions of
*r* and *theta*. So cylindrical coordinates are a good
choice. Note that spherical coordinates would not be as nice for
this problem. Check that you can see why not (what boundary(ies)
isn't (aren't) nicely described by spherical coordinates?).

*(How were the figures here generated? In Maple, with this
maple worksheet.)*

cylindrical coordinates

Last modified: Mon Mar 15 11:00:16 EST 2004

Comments to:glarose@umich.edu

©2004 Gavin LaRose, UM Math Dept.