**Double Integrals in Maple**

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

Maple can help you determine the appropriate limits of integration for evaluating a double integral
*
*
(1)

over a region
*D*
in the
*xy*
-plane. In (1) the "Type I" region
*D*
lies between the graphs of two continuous functions
and
of
*x*
over the interval [
*a*
,
*b*
], where the graph of
lies below that of
throughout
*D.*
Imagine shooting an arrow upward through the region. The lower curveÑwhere the arrow first pierces
*D*
Ñprovides the formula for the lower limit in the first integration, which takes place with respect to
*y.*
The upper curveÑwhere the arrow leaves
*D*
Ñgives the formula for the corresponding upper limit.

Exercise 14, Section 16.3 of James Stewart,
*Calculus, 4th Edition*
, ITP Brooks/Cole, 1999 illustrates (1). In using the Maple routine below for other regions, change the functions in the
region :=
line and experiment with the parameters there and also in the
arrow
command to size that appropriately.

**Example.**
Evaluate (1) for
*f*
(
*x*
,
*y*
) =
*x*
+
*y*
over the region between the graphs of
*y*
=
and
*y*
=
.

**Solution.**
The following routine plots the region with an upward-directed arrow through it.

`> `
**with(plots):
plotsetup(inline);
with (plottools):
region := plot( [sqrt(x), x^2], x = -1..1.5, color = red ):
regname := textplot([.25, .35, "D"], font = [TIMES, ITALIC, 14], color = red):
slice := arrow([.5, -.5], [.5, 1.5], .01, .12, .125, color = blue):
display(region, slice, regname, labels = ["x", "y"] );**

From the routine's output figure, the lower limit of integration is the
*x*
-axis:
*y*
= 0. The upper limit coresponds to the top parabola, whose equation is
*y*
=
Thus,

(2)
=

You can use Maple to check hand evaluation of (2), which gives 3/10. Maple's
student
package contains the command
DoubleInt
to evaluate double integrals by iteration:

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

It is often preferable to evaluate a double integral by iteration in which the first integration is with respect to the variable
*x*
instead of
*y*
. A "Type II" region
*D*
lies between the graphs of two continuous function
and
of
*y*
over an interval [
*c*
,
*d*
], where the graph of
lies to the left of the graph of
throughout
*D. *
For such
* D, *
evaluation of the double integral by iteration takes the form

(3)
* *
* = *

To determine whether (1) or (3) is better for a specific double integral, set up both iterated integrals. As an illustration, consider the region
*D*
of integration for Exercise 24, Section 16.3, of Stewart's
*Calculus*
.
*D*
lies in the first quadrant between the graphs of
*y*
=
2 and
*x*
= 2
*y*
.

**Example 2**
. Set up (1) for this region
*D*
if
*f*
(
*x*
,
*y*
) =
.

**Solution**
. The following routine plots
*D*
with an upward-directed arrow through it.

`> `
**with(plots):
plotsetup(inline);
with (plottools):
region := plot( [2, x/2], x = 0..4, color = red ):
regname := textplot([1, 1, "D"], font = [TIMES, ITALIC, 14], color = red):
slice := arrow([1.8, -.2], [1.8, 2.5], .05, .2, .1, color = blue):
display(region, slice, regname, labels = ["x", "y"]);**

From this figure, Equation (1) gives

=

Integrating first with respect to
*y*
is not appealing, so it is worthwhile to set up (3). The next routine plots
*D*
with a left-to-right arrow through it.

`> `
**with(plots):
with (plottools):
region := plot( [2, x/2], x = -.2..4.2, color = red ):
regname := textplot([1, 1, "D"], font = [TIMES, ITALIC, 14], color = red):
slice := arrow([-.2, 1.2], [3.75, 1.2], .04, .15, .1, color = blue):
display( region, slice, regname, labels = ["x", "y"] );**

From this figure the limits of integration in (3) are clear:

=

As you should confirm, hand evaluation of this iterated integral gives
. The calculation is intricate enough to have Maple check it:

`> `
**with (student):
value(Doubleint(sqrt(4 - y^2), x = 0..2*y, y = 0..2));**

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