15 Multiple Integrals Copyright Cengage Learning All rights reserved 15 4 Double Integrals in Polar Coordinates Copyright Cengage Learning All rights reserved 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 3 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 4 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 Polar rectangle Figure 3 Figure 1 5 Double Integrals in Polar Coordinates In order to compute the double integral R f x y dA where R is a polar rectangle we divide the interval a b into m subintervals ri 1 ri of equal width r b a m and we divide the interval into n subintervals i 1 j of equal width n Then the circles r ri and the rays j divide the polar rectangle R into the small polar rectangles Rij shown in Figure 4 Dividing R into polar subrectangles Figure 4 6 Double Integrals in Polar Coordinates The center of the polar subrectangle Rij r ri 1 r ri i 1 j has polar coordinates We compute the area of Rij using the fact that the area of a sector of a circle with radius r and central angle is 7 Double Integrals in Polar Coordinates Subtracting the areas of two such sectors each of which has central angle j j 1 we find that the area of Rij is Although we have defined the double integral R f x y dA in terms of ordinary rectangles it can be shown that for continuous functions f we always obtain the same answer using polar rectangles 8 Double Integrals in Polar Coordinates The rectangular coordinates of the center of Rij are so a typical Riemann sum is If we write g r rf r cos r sin then the Riemann sum in Equation 1 can be written as which is a Riemann sum for the double integral 9 Double Integrals in Polar Coordinates Therefore we have 10 Double Integrals in Polar Coordinates The formula in says that we convert from rectangular to polar coordinates in a double integral by writing x r cos and y r sin using the appropriate limits of integration for r and and replacing dA by r dr d Be careful not to forget the additional factor r on the right side of Formula 2 A classical method for remembering this is shown in Figure 5 where the infinitesimal polar rectangle can be thought of as an ordinary rectangle with dimensions r d and dr and therefore has area dA r dr d Figure 5 11 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 by 1 r 2 0 Figure 1 b 12 Example 1 Solution cont d Therefore by Formula 2 13 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 Figure 7 where D is a type II region we obtain the following formula 14 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 15
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