Double Integrals in Maple

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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