Math 2E 44275 Spring 2014 Quiz 6 5 29 14 Show all of your work Partial credit will be given 1 10 points Evaluate the surface integral Name Print ID Number ZZ F dS S for F x y z xzi xj yk and S is the hemisphere x2 y2 z2 4 y 0 oriented in the direction of the positive y axis Solution Using spherical coordinates S is given by Therefore and so Then ZZ S F dS x 2 sin cos y 2 sin sin z 2 cos 0 0 F r 2 sin cos 2 cos i 2 sin cos j 2 sin sin k r r 4 sin2 cos i 4 sin2 sin j 4 cos sin k F r r r 16 sin3 cos cos2 8 sin3 cos sin 8 sin2 cos sin 16 sin3 cos cos2 8 sin3 cos sin 8 sin2 cos sin d d D F r r r dA ZZ Z Z cid 20 Z Z cid 0 sin3 cos 2 sin2 cos cid 1 d 8 2 sin3 cos cid 18 1 2 sin 2 cid 19 1 4 0 0 0 sin3 8 8 0 cid 18 1 cid 20 2 1 4 cid 19 sin2 sin4 2 1 3 cid 21 0 sin3 0 cid 21 2 0 sin2 cos cos d
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