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Discussion Set C5: Cylindrical and Spherical Coordinates
© 1994 by CaRP, Department of Mathematics, University of California, Davis

The purpose of this discussion set is to improve your understanding of and to help you strengthen your skills in writing integrals over regions in space using cylindrical and spherical coordinates. Use of these coordinate systems can often simplify integrals that are difficult to evaluate in rectangular coordinates.

Self-Help Background Check

In order to get the most benefit from the discussion period, do the following problems before the start of discussion. You can check your answers with those given after the end of the discussion assignment.

Below, draw the Cartesian coordinates $(x,y)$ , and on the same graph draw the polar coordinates $r$ and $h$ .
Find equations for $x$ and $y$ in terms of the polar coordinates $r$ and $h$ .
Convert each of the following equations into equations involving polar coordinates:
1. $x2 + y2 = 4$
2. $x = 1$
3. $xy = x + y$

If you missed more than one problem, consult our text on page 519, or if time permits, go to office hours for clarification. In addition, problems 9 and 11 on page 525 provide extra practice.

Guided Inquiry

These guidelines will help you to know when to use a coordinate system other than rectangular coordinates.

Cylindrical coordinates are particularly useful when the region of integration involves circles (in the $xy$ plane) and when $z$ can be written as a function of $x$ and $y$ that is not too complicated. Cylindrical coordinates can be thought of as polar coordinates with a vertical dimension appended. In cylindrical coordinates,

integral integral integral integral R f(P )dV = f (r,h,z)rdz drdh,

where $dV = r dz drdh$ .

To convert from cylindrical to rectangular coordinates, note that

x = r cosh, y = rsinh, z = z.

In addition, $x2 + y2 = r2$ , just as in polar coordinates.

The figure below illustrates the relationship between rectangular and cylindrical coordinates. Notice that $x$ , $y$ , $r$ , and $h$ are related just as they are in polar coordinates,while $z$ is the same as in rectangular coordinates.

Note: The sample unit that is helpful in setting up the limits of integration in cylindrical coordinates looks like a wedge that has been cut out of a washer – see Figure 10 on p. 920 of our text.

Describe and sketch the graph of each of the following equations in cylindrical coordinates.
1. $r = 3$
2. $p h = 4$

Spherical coordinates

are particularly useful when the region of integration involves spheres and cones. Spherical coordinates consist of two coordinates that describe angles and one that describes the distance from the pole to a point $P$ in space. In spherical coordinates,

integral integral integral integral f (P)dV = f(r,h, f)r2sinf dr df dh, R

where $2 dV = r sin f drdf dh$ . To convert from spherical to rectangular coordinates, note that

x = r sin fcos h, y = rsinf sin h, z = rcos f

expresses the relationship between variables in the two coordinate systems.

The first figure illustrates the spherical coordinate system. The second figure illustrates the relationship between rectangular and spherical coordinates.

Notice that

r = r sin f.

Using the relationship $x = rcosh$ and $y = r sin h$ , we obtain

x	= sin cos	(1)
y	= sin sin	(2)
z	= cos .	(3)

Note: The sample unit that is helpful in setting up the limits of integration in spherical coordinates looks like a box whose edges got a bit warped – see Figure 10 on p. 924 of our text. Alternatively, you can think of it as a chunk of an orange peel.

Describe and sketch the graph of each of the following equations in spherical coordinates. You may need to use the equations given above for converting from spherical to rectangular coordinates.
1. $p f = -- 4$
2. $r = 5csc f$

Finally, comparable to setting up an integral in rectangular coordinates, when writing an integral in cylindrical or spherical coordinates, the limits of integration cannot involve a variable with respect to which an integration has already been performed. For example, if $dz$ is the innermost differential, then the outside and middle limits of integration cannot involve the variable $z$ .

The region is below the sphere and above the cone
1. Sketch a graph of $R$ .
2. Find the equation of the level curve where the sphere and cone intersect. Sketch its projection in the $xy$ (i.e., the $z = 0$ ) plane.
3. Convert the equations of the surfaces that bound $R$ into cylindrical coordinates.
4. Fill in what’s missing to write an integral for the mass of $R$ , if the density at a point within $R$ is twice the distance from the point to the $z$ axis.
Recall that mass $integral = d(P )dV$ , where $d(P )$ is the density at the point P. First, note that we can write the desired integral either by writing it first in rectangular coordinates and converting it to cylindrical coordinates, or by “thinking cylindrically” and writing the integral directly. Let’s use the latter approach here.
Because the density at the point $(x, y,z)$ within $R$ is twice the distance from $(x,y,z)$ to the $z$ axis, we can write an expression for density as
$d(P ) = 2r.$
To describe the region $R$ in cylindrical coordinates, it is usually easiest to fix $r$ and/or $h$ , then write $z$ as a function of $r$ and $h$ . Hence, from the projection on the $z = 0$ plane,
$0 < r < ----- and ----< h < 2p.$
Now the $z$ coordinate extends from the cone up to the hemisphere. Thus, $z$ must lie between the equations for the two surfaces. Solving for $z$ in these equations, we obtain
$V~ -------- z = 36 - for the sphere, and ---$

$z = -----for the cone.$

Therefore, the inequality that describes the range for $z$ is
$V~ - 3r < z < ----.$
The integral in cylindrical coordinates becomes
$integral integral 2p integral V~ 36-- --- --- 2rr dz dhdr. 0 r --- ---$

Group Investigation

Let’s turn to spherical coordinates and do some more work with the region that was given in (6).

The region is bounded above by the sphere and below by the cone .
1. Describe $R$ using spherical coordinates. Check your answer with your discussion facilitator before going on.
2. Set up an integral to find the volume of $R$ using spherical coordinates.

Discussion Set C5 Assignment

At the start of class on the day indicated by your instructor, hand in your well-written work for the following problems, Refer to the “Mathematical Writing” section in Chapter 1 of Cameos for guidance on style. Feel free to discuss the problems with others, BUT your solutions should be written independently in your own words.

For the following problems, set up an integral to describe the desired quantity. For each problem, draw the region, choose a coordinate system (rectangular, cylindrical, or spherical) so that the region is easy to describe and the integral is easy to compute. Also show the sample unit related to the coordinate system that you are using. Then, write a paragraph explaining your choice of coordinate system, and the consequences of using other coordinate systems.

Set up an integral for the volume of the solid that lies above the $xy$ -plane, below the plane $x + z = 5$ and inside the cylinder $2 2 x + y = 9$ .
Set up an integral for the mass of $R$ , where $R$ is the region bounded above by the sphere $x2 + y2 + z2 = 36$ and bounded below by the cone $V~ - V~ --2---2- z = 3 x + y$ , and the density at a point $P$ within $R$ is the distance from $P$ to the point $(0, 0,0)$ .
Suppose you wish to compute $integral f(P )dV , R$
where $R$ is the solid bounded between the paraboloid $z = 6 - (x2 + y2)$ and the cone $-------- z = V~ 3V ~ x2 + y2$ . Without knowing $f(P )$ , discuss the advantages and disadvantages of using
1. cylindrical
2. spherical
coordinates for this computation.

Brief answers to the problems from the Self-Help Background Check:

(1a)

(2) $x = rcos h$ , and $y = r sin h$

(3a) $r2 = 2$

(3b) $r = sec h$

(3c) $r = sec h + csch$

Mathstory

Richard Courant (1888 - 1972) was a mathematician who worked in function theory and a branch of applied mathematics called the calculus of variations. He received his doctorate from the University of Göttingen in 1910, and later became founder and director of its institute of mathematics. During that time, he became close friends with David Hilbert and coauthored a two-volume treatise, Methods of Mathematical Physics, that today remains a classic and important reference work. Courant,like many of his peers, fled Germany in 1933. He became a professor of mathematics at New York University and was one of the founders of a mathematical research institute at NYU, which was named after him posthumously.

Literature Cited

Encyclopedia Brittanica, 1989. Vol 3. Chicago, IL.

Discussion Set C5: Cylindrical and Spherical Coordinates © 1994 by CaRP, Department of Mathematics, University of California, Davis