13 Notes 10

13 Notes 10 - RR D xydA, where D : x 1 ,y , ( x-1) 2 + y 2...

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taught by J. Lebl Here is a list of the examples explained by Prof. Lebl. I am not going to write down the solutions, since I think it would be a good exercise for everyone to go through them and compute on their own. The one exception is Example 3, because this is a novel way of using double integrals. The multivariable techniques are used to compute a single variable integral which could not be computed with only single variable calculus knowledge. For an exposition of polar and cylindrical coordinates, see the notes for week 10. Example 1 Integrate xy + y 2 over the region in plane described in polar coordinates by 1 r 2 , - π/ 2 θ π/ 2 . This is a half annulus. In polar coordinates, xy + y 2 = r 2 cos θ sin θ + r 2 sin 2 θ. Example 2
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Unformatted text preview: RR D xydA, where D : x 1 ,y , ( x-1) 2 + y 2 1 . Example 3 R - e-x 2 dx. Denote by A our integral. It will be non-negative since the exponential is positive. Then A 2 = A A = Z - e-x 2 dx Z - e-y 2 dy = Z - Z - e-x 2-y 2 dxdy. Changing to polar coordinates, this gives A 2 = Z 2 Z re-r 2 dr. The inner integral is equal, via the change of variables u = r 2 , to 1 2 Z e-u du = 1 2 . Hence A 2 = , and A = . Example 4 Let W : x 2 + y 2 1 , z 1 + x 2 + y 2 . ZZZ W ( x 2 + y 2-z ) dV = Z 2 Z 1 Z 1+ r 2 ( r 2-z ) rdzdrd = ... Example 5 The volume of the unit ball B in R 3 can be computed using cylindrical coordinates (4 / 3). 1...
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