Example 1
Compute the integral
Solution: We will compute the double integrals as the iterated integral
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To finish, we need to compute the integral with respect to y, which is simple. Since x is gone, it's just a regular one-variable integral. We calculate that our double integral is
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To double check, we can compute the integral in the other direction, integrating first with respect to y and then with respect to x. The only trick is to remember that when integrating with respect to y, we must think of x as a constant. Since the integral of cy2 is cy3/3, we calculate
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Example 2
Rectangular regions were easy because the limits
(
a
x
b and
c
y
d) were fixed. Here's an example
where we integrate over the triangle defined by
0
x
2,
0
y
x/2.
Use same function
f (x, y) = xy2. Compute
xy2dA
when D is that triangle.
Solution: Note for the triangle defined by
0
x
2
and
0
y
x/2, the limits on y depend on x. For a given
value of x, y ranges from 0 to x/2, as illustrated above by the
vertical dashed line from (x, 0) to (x, x/2).
In a double integral, the outer limits must be constant, but the inner limits can depend on the outer variable. This means, we must put y as the inner integration variables, as was done in the second way of computing Example 1. The only difference from Example 1 is that the upper limit of y is x/2. The double integral is
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Example 2'
Now compute the integral over the same triangle D, but make y be the outer integration variable.
Solution: Now we need to give constant limits for y. As
illustrated below, for point in the triangle y ranges between
between 0 and 1. Then, for a given value of y, x takes on
values between 2y and 2 (as shown by the horizontal dashed line
between (2y, y) and (2, y)). Hence, we can describe the triangle by
0
y
1 and
2y
x
2.
Is it confusing that the limits of x are
2y
x
2 rather
than
0
x
2 (which would more closely parallel the above
Example 2)? If we let x range from 0 to 2y, then the triangle
would be the upper-left triangle in the above picture. We want to
compute the integral over the region D, which is the lower-right
triangle shaded in red. In this triangle, y < x/2 (as used above in
Example 2) which means that x > 2y (as we are now using in Example 2').
The double integral is similar to the first way of computing Example 1, with the only difference being that the lower limit of x is 2y. The integral is
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To go from Example 2 to Example 2', we "changed the order of integration." You can see more examples of changing the order of integration in double integrals here.