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.