25-Polar Regions

25-Polar Regions - Double Integrals Polar Coordinates John...

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Unformatted text preview: Double Integrals: Polar Coordinates John E. Gilbert, Heather Van Ligten, and Benni Goetz The Astrodome in Houston might be modelled mathematically as the region below a sphere x 2 + y 2 + z 2 = R 2 and above a circular disk D = { ( x, y ) : x 2 + y 2 ≤ a 2 } . It’s natural to ask for its volume. In terms of double integrals this is Volume = D R 2- x 2- y 2 dxdy , and the integral probably could be evaluated with some effort. But since everything is rotationally symmetric perhaps we should try changing to polar coordinates ( r, θ ) . First let’s see why we should think about changing variables ( x, y )-→ ( r, θ ) , x = r cos θ , y = r sin θ from Cartesian to polar coordinates. An example makes things clearer. Example 1: Suppose D is the semicircular region shown to the right and that I = D f ( x, y ) dxdy for some unspecified f . θ r x y 2- 2 Before to evaluate I we’d probably have first fixed x and integrated with respect to y along the vertical black line. Then in Cartesian coordinates,vertical black line....
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This note was uploaded on 02/13/2012 for the course M 408 D taught by Professor Textbookanswers during the Spring '07 term at University of Texas.

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25-Polar Regions - Double Integrals Polar Coordinates John...

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