**Triple Integrals and Cylindrical Coordinates**

**Copyright © 1999, 2001 by James F. Hurley, University of Connecticut Department of Mathematics, Unit 3009, Storrs CT 06269-3009. All rights reserved.**

**1. Basic surfaces and basic cylidindrical-coordinate regions**
.
Recall Maple's command for plotting surfaces described by a cylindrical-coordinate equation of the form
*r*
=
*f*
(
,
*z*
):

cylinderplot( f( , z), = .. , z = c..d ) ;

It is effective for surfaces lying above a region of the
*xy*
-plane whose polar-coordinate description is a basic "polar rectangle" [
,
]
´
[
*c*
,
*d*
]. This is enough for some types of triple-integration problems.

**Example 1**
. Find the volume (for example, of ice cream) above the graph of
*z*
=
and below the sphere with equation
= 4.

**Solution**
. The first step is to plot the region
*E*
between the cone and the sphere. The following routine does that. In coding the routine, the first step was determining algebraically the curve
*C*
of intersection of the cone and the sphere.
*C*
is also the boundary of the subregion
*D*
of integration in the
*xy*
-plane. To get the equation of
*C*
, substitute
=
from the first equation into the second:

= 4 Þ ,

which is the circle of radius
centered at the origin. In polar coordinates that is
*r*
=
.
Note that the equation of the cone transforms to the cylindrical-coordinate equation
*z*
=
*r*
,
and the equation of the sphere transforms to
= 4. Since all the ice cream lies above the cone Ñ in particular, above the
*xy-*
plane Ñ in solving the equation of the sphere for
*r*
only the positive square root is necessary:
*r*
=
. Letting
range just from 0 to ¹ reveals the interior the region, from which it easy to determine the proper limits of integration. Rotation of the initial plot gives a good image of the region
*E*
. Execute the routine, and then experiment with rotating it until you have a good view of the region.

`> `
**with (plots):
surf1 := cylinderplot (z, theta = 0..Pi, z = 0..sqrt(2), axes = boxed):
surf2 := cylinderplot (sqrt(4 - z^2), theta = 0..Pi, z = 0..2):
xaxis := spacecurve([t, 0, 0, t = -2..2, color = magenta]) :
yaxis := spacecurve([0, t, 0, t = -2..2, color = magenta]) :
zaxis := spacecurve([0, 0, t, t = -1..3, color = magenta]) :
labx := textplot3d([2.1, 0, -.2, `x`], color = magenta):
laby := textplot3d([0,2.1, -.2, `y`], color = magenta):
labz := textplot3d([0, 0, 3.1, `z`], color = magenta):
display(surf1, surf2, xaxis, yaxis, zaxis, labx, laby,labz);**

An arrow shot upward through
*E*
enters the region by piercing the cone
*z*
=
*r*
and exits by emerging from the top of the sphere
*z*
=
.
Bearing in mind that
*dV*
=
*r dr*
*d*
*d*
*z*
,
the volume is then

* *
=
=

=
=
.

As a check, execute Maple's built-in
TripleInt
command from the
student
package. Keep in mind that, just as for the command is designed only for Cartesian-coordinate triple integrals. So you must give it the factor
*r*
in
*dV*
=
*r*
*dr*
*d*
*dz*
as part of the integrand!

`> `
**with (student):
value(Tripleint(r, z = r..sqrt(4 - r^2), r = 0..sqrt(2), theta = 0..2*Pi) );**

**2. More general surfaces and regions**
.
Maple's
cylinderplot
* *
command can plot any cylindrical-coordinate equation for which
*z*
or
is expressible as a function of the remaining two variables. Recall that the syntax of the extended
cylinderplot
command requires entry of the formulas for all three of
*r*
,
, and
*z*
(in that order),
*and*
specification of the range of the two independent ones, for which we use the generic symbols
*u*
and
*v:
*

cylinderplot( [r, , z], u = a..b, v = c..d ) ;

For example, the next routine plots the two simple cylindrical surfaces with equations
*z*
=
*c*
and
=
. (In the second plot, if you change to multiples of ¹, the initial rendering appears edge-on, so click and rotate the plot slightly to see the plane.)

`> `
**with (plots):
cylinderplot( [r, theta, 3], r = 0..2, theta = 0..2*Pi, axes = boxed );
cylinderplot( [r, 2, z], r = 0..2, z = 0..3, axes = boxed );**

*
*
The concluding example of this worksheet involves a triple integral over a region whose bounding surfaces are plotted by the extended
cylinderplot
command. It is representative of a common way that cylindrical-coordinate triple integrals arise: by transformation of a complicated Cartesian-coordinate triple integral.

`> `
**with (plots):
surf1 := cylinderplot (2*cos(theta), theta = 0..Pi/2, z = 0..2, axes = boxed):
surf2 := cylinderplot ( [r, theta, r], r = 0..2, theta = 0..Pi/2):
xaxis := spacecurve([t, 0, 0, t = 0..2, color = magenta]) :
yaxis := spacecurve([0, t, 0, t = 0..2, color = magenta]) :
zaxis := spacecurve([0, 0, t, t = 0..2.5, color = magenta]) :
labx := textplot3d([2.1, 0, -.2, `x`], color = magenta):
laby := textplot3d([0, 2.1, -.2, `y`], color = magenta):
labz := textplot3d([0, .2, 2.2, `z`], color = magenta):
display(surf1, surf2, xaxis, yaxis, zaxis, labx, laby,labz);**

From the figure, the roof of the region is the conical surface, which an upward-directed arrow through
*E*
hits as it emerges from the region, after entering
*E*
through the
*xy*
-plane. Thus, since
*dV*
=
*r dr d*
*dz*

=
,

where
*D*
is the half disk in the
*xy*
-plane with bounding curve
*r*
= 2 cos
as
varies from 0 to ¹/2. Thus,

=
=
.

Evaluate the last integral by means of the trigonometric identity
:

=
=
.

As before, you can check the entire calculation by using Maple's
TripleInt
command from the
student
package:

`> `
**with (student):
value( Tripleint(1/sqrt(x^2 + y^2), z = 0..sqrt(x^2 + y^2), y = 0..sqrt(2 - x^2), x = 0..sqrt(2)) );**