Copyright © Cengage Learning. All rights reserved. 15 Multiple IntegralsCopyright © Cengage Learning. All rights reserved. 15.4 Double Integrals in Polar Coordinates3 3 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 solids4 4 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 15 5 Double Integrals in Polar Coordinates Recall from Figure 2 that the polar coordinates (r, θ ) of a point are related to the rectangular coordinates (x, y) by the equations Figure 26 6 Double Integrals in Polar Coordinates The regions in Figure 1 are special cases of a polar rectangle R = {(r, θ ) | a ≤ r ≤ b, α ≤ θ ≤ β } which is shown in Figure 3. Figure 1 Polar rectangle Figure 37 7 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 Δr = (b – a)/m and we divide the interval [α, β ] into n subintervals [θi – 1, θj] of equal width Δθ = (β – α)/n. Then the circles r = ri and the rays θ = θj divide the polar rectangle R into the small polar rectangles Rij shown in Figure 4. Figure 4 Dividing R into polar subrectangles8 8 Double Integrals in Polar Coordinates The “center” of the polar subrectangle Rij = {(r, θ ) | ri – 1 ≤ r ≤ ri, θi – 1 ≤ θ ≤ θj} 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 θ is .9 9 Double Integrals in Polar Coordinates Subtracting the areas of two such sectors, each of which has central angle Δθ = θj – θj – 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.10 10 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, θ ) = r f (r cos θ, r sin θ ), then the Riemann sum in Equation 1 can be written as which is a Riemann sum for the double integral11 11 Double Integrals in Polar Coordinates Therefore we have12 12 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 θ and y = r sin θ, using the appropriate limits of integration for r and θ, and replacing dA by r dr dθ. 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 dθ and dr and therefore has “area” dA = r dr dθ. Figure 513 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 where D is a type II region, we obtain the following formula. Figure
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