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

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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 e ff ort. 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.
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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, I = 2 - 2 4 - x 2 0 f ( x, y ) dy dx, which might not be easy to evaluate. But why might integrating instead along the red lines be easier?
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