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 Learning objectives convert double integrals from rectangular coordinates to polar coordinates evaluate double integrals over polar rectangles and polar regions of type II determine the region of integration that is bounded by two or more plane curves in polar coordinates use polar coordinates to find areas of regions and volumes of solids 3 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 4 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 5 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 6 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 7 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 8 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 9 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 r f r cos r sin then the Riemann sum in Equation 1 can be written as which is a Riemann sum for the double integral 10 Double Integrals in Polar Coordinates Therefore we have 11 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 12 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 13
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