15.4 - Section 16.4 Double Integrals in Polar Coordinates...

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Unformatted text preview: Section 16.4 Double Integrals in Polar Coordinates “Integrating Functions over circular regions” Suppose we want to integrate the function f ( x, y ) = x 2 over the fol- lowing region: 3 2 We would have to break the region up into three pieces. On the first region we would have- 2 lessorequalslant x lessorequalslant 2 and √ 4- x 2 lessorequalslant y lessorequalslant √ 9- x 2 , on the second region- 3 lessorequalslant x lessorequalslant 2 and 0 lessorequalslant y lessorequalslant √ 9- x 2 and on the last region 2 lessorequalslant x lessorequalslant 3 and 0 lessorequalslant y lessorequalslant √ 9- x 2 . Thus we would have to evaluate the following integrals: integraldisplay- 2- 3 integraldisplay √ 9- x 2 x 2 dydx + integraldisplay 2- 2 integraldisplay √ 9- x 2 √ 4- x 2 x 2 dydx + integraldisplay 3 2 integraldisplay √ 9- x 2 x 2 dydx. Not only was the process of setting up this integral laborious, but to evaluate the first integral, integraldisplay- 2- 3 integraldisplay √ 9- x 2 x 2 dydx = integraldisplay- 2- 3 x 2 √ 9- x 2 dx we would need to use a trigonometric substitution, which is itself a difficult and time consuming task. Therefore, we need to develop new techniques of integration to help us integrate functions over different types of regions. Recall in Calculus 2 that when considering problems over circular re- gions, instead of using Cartesian coordinates, we can instead use polar coordinates to try to determine an integral. In this section we shall determine how to use polar coordinates to evaluate an integral. 1 2 1. Evaluating Integrals in Polar Coordinates over Polar Rectangles . Suppose that f ( x, y ) is continuous functions and we want to find the integral integraltext integraltext D f ( x, y ) dA over some region D as illustrated below. Then instead of integrating in Cartesian coordinates, we can use in polar coordinates. In order to determine exactly how, we shall first consider the problem over a polar rectangle....
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