11.04 - Section 12.7: Spherical coordinates Main ideas?...

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Unformatted text preview: Section 12.7: Spherical coordinates Main ideas? ..?.. Spherical coordinates: (x,y,z)Cartesian = (,,)polar, where x = sin cos , y = sin sin , z = cos . with 0, 0 2, 0 . Triple integrals can be easier to evaluate in spherical coordinates if the domain of integration has spherical symmetry, and/or the integrand has spherical symmetry. If E = {(,,)spherical : a b, , c d}, then ..?.. a b E f(x,y,z) dV = ... f( sin cos , sin sin , cos ) 2 sin d c d d d. (Stewart writes it differently: he writes d before d.) Compute volume of upper unit hemisphere using spherical coordinates: ..?.. E = {(,,)spherical : 0 1, 0 2, 0 /2 }, so volume(E) = E 1 dV = ... ..?.. 01 02 0 /2 2 sin d d d = ... ..?.. ( 01 2 d) ( 02 1 d) ( 0 /2 sin d) = ((1/3) 3 |01) (2) (cos |0 /2) = (1/3) (2) (1) = 2/3 just like last time. ...
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11.04 - Section 12.7: Spherical coordinates Main ideas?...

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