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Unformatted text preview: Physics 31 Spring, 2008 Solution to HW #11 Problem C Evaluate the following integrals. You should write the cartesian coordinates x or z in terms of spheri- cal polar coordinates. Remember that the volume element will be r 2 dr sin d d , and the integrals are over all space. Rearrange terms so that each integral is a product of an r integral, a integral, and a integral. It usually pays to evaluate the and integrals first, because if one of them is zero, you have the answer without bothering with the r integral. For all of these integrals, Z . . . dV = Z r 2 dr Z sin d Z 2 d . . . , z = r cos , and x = r sin cos . (a) Z 100 z 100 dV = 1 a 3 Z e 2 r/a r cos dV For this one, the integral is zero: Z cos sin d = 1 2 cos 2 = 0 . This integral can also be evalutated by the substitution u = cos , which is always a good general approach....
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