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L13_5 - Spring 2004 Math 253/501503 13 Multiple Integrals...

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Spring 2004 Math 253/501–503 13 Multiple Integrals 13.5 Double Integrals in Polar Coordinates Thu, 26/Feb c 2004, Art Belmonte Summary According to which type of region we have, the rectangular double integral ZZ D f ( x , y ) dA is realized as one or both of the following iterated polar integrals. Z β α Z h 2 (θ) h 1 (θ) f ( r cos θ, r sin θ) r dr d θ Z b a Z g 2 ( r ) g 1 ( r ) f ( r cos θ, r sin θ) r d θ dr The first of these is by far the most common. Note Learn your area differentials! dA = dx dy = dy dx = r dr d θ Don’t forget the JacFac! (That is, remember to include the multiplicative factor r in the area differential when converting to polar coordinates. We’ll say more about this in Section 13.11.) Hand / MATLAB Examples Set up a problem by hand, then use your TI-89 or MATLAB to dispatch it. Compute a multiple integral directly. Give both an exact value as well as an approximation—to get a feel for the magnitude of the answer. (If you’d like to see the steps involved, just use smi . Of course, it is illustrative to do a few problems
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