**Double Integrals and Polar Coordinates**

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

1. Plots via
**polarplot**
. Maple's
plots
library includes the command
polarplot
, which directly plots curves with polar-coordinate equations
*r*
=
*f*
(
)
. (It can also plot polar-coordinate equations parametrically: refer to Maple's
*Learning Guide*
or on-line
Help
.) The syntax of the
polarplot
command is

polarplot( f( ), = .. ) ;

The following routine illustrates it for the circle
*r*
= 4.
* *
Note inclusion of the command
scaling = constrained
to make the circle appear circular (rather than elliptical). Also note that the default plot shows the
*x*
- and
*y*
-axes. That is handy, because in most cases polar coordinates arise in double integrals when a given
*xy*
-region is simpler to describe and integrate over in polar coordinates. Execute the routine by placing your cursor at the end and hitting the Enter key.

`> `
**with (plots):
polarplot (4, theta = 0..2*Pi, scaling = constrained);**

**2. Polar Coordinates and Double Integrals**
. Exercise 12 from Section 16.4 of James Stewart,
*Calculus, 4th Edition*
, ITP Brooks/Cole, 1999, illustrates the simplification that change to polar coordinates can bring to evaluation of double integrals.

**Example 1**
. Evaluate the double integral of
*f*
(
*x*
,
*y*
) over the disk
² 16 if

.

**Solution**
. The formula for the integrand is quite involved, but changing to polar coordinates transforms it to a much simpler form in which the denominator becomes (
)
to the power 3/2. Hand evaluation of the resulting integral

** **

gives
,
which you can ask Maple to check with its
Doubleint
command in the
student
package.

`> `
**with (student):
value( Doubleint( 1/(1 + r^2)^(3/2), r = 0..4, theta = 0..2*Pi) );**

Oops! What went wrong? The hand calculation? No, actually the
Doubleint
command has a serious limitation: it works only for
*Cartesian-coordinate integrals*
! So you must manually supply the factor
*r*
in the formula

* dA*
=
*r*
*dr*
*d*

and have Maple evaluate the iterated integral as in the routine below.

`> `
**with (student):
value(Doubleint(r/(1 + r^2)^(3/2), r = 0..4, theta = 0..2*Pi) );**

As you should confirm,
*that*
agrees with the hand calculation.