copyright © cengage learning. all rights reserved. 15.4 double integrals in polar coordinates
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15.4 Double Integrals in Polar Coordinates
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Double Integrals in Polar Coordinates
Suppose that we want to evaluate a double integral R f (x, y) dA, where R is one of the regions shown in
Figure 1. In either case the description of R in terms of rectangular coordinates is rather complicated, but R is easily described using polar coordinates.
Figure 1
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Double Integrals in Polar Coordinates
Recall from Figure 2 that the polar coordinates (r, ) of a point are related to the rectangular coordinates (x, y) by the equations
Figure 2
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Double Integrals in Polar Coordinates
The regions in Figure 1 are special cases of a polar rectangle
R = {(r, ) | a r b, }
which is shown in Figure 3.
Figure 1
Polar rectangle
Figure 3
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Example 1
Evaluate R (3x + 4y2) dA, where R is the region in the upper half-plane bounded by the circles x2 + y2 = 1 and x2 + y2 = 4.
Solution:
The region R can be described as
R = {(x, y) | y 0, 1 x2 + y2 4}
It is the half-ring shown in Figure 1(b),and in polar coordinates it is given by1 r 2, 0 . Figure 1(b)
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Example 1 – Solutioncont’d
Therefore, by Formula 2,
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Double Integrals in Polar Coordinates
What we have done so far can be extended to the more complicated type of region shown in Figure 7.In fact, by combining Formula 2 with
where D is a type II region, we obtain the following formula.
Figure 7
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Double Integrals in Polar Coordinates
In particular, taking f (x, y) = 1, h1( ) = 0, and h2( ) = h( ) in
this formula, we see that the area of the region D bounded by = , = , and r = h( ) is
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