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Unformatted text preview: disk, we have D  x  dA = L  x  dA + R  x  dA . But we also have L  x  dA = R  x  dA (if this isnt obvious to you, think about Riemann sums). So M = 4 R  x  dA = 4 R x dA . Lets do this two ways: Cartesian coordinates: R x dA = ..?. . 1 sqrt(1 xx ) sqrt(1 xx ) x dy dx = 1 2 x sqrt(1 x 2 ) dx = (2/3)(1 x 2 ) 3/2  1 = 2/3 . So M = 4(2/3) = 8/3. Polar coordinates: R x dA = ..?. . /2 /2 1 ( r cos ) r dr d = ..?. . ( /2 /2 cos d ) ( 1 r 2 dr ) = (sin  /2 /2 ) ( r 3 /3  1 ) = (2)(1/3) = 2/3 as before. Proceed as above....
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 Fall '11
 Staff
 Integrals, Polar Coordinates

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