practice32answers - Practice Exam 2 Solutions The answer to...

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Practice Exam 2 Solutions The answer to evey multiple choice question is a . 9. Under the change of variables x = s 2 - t 2 ,y = 2 st the quarter circular region in the st -plane given by s 2 + t 2 1 ,s 0 ,t 0 is mapped onto a certain region D of the xy -plane. Evaluate ZZ D dxdy p x 2 + y 2 . To evaluate this integral, we note that ( x,y ) ( s,t ) = ± ± ± ± 2 s - 2 t 2 t 2 s ± ± ± ± = 4 s 2 - ( - 4 t 2 ) = 4 s 2 + 4 t 2 > 0 and 1 p x 2 + y 2 = 1 p ( s 2 - t 2 ) 2 + (2 st ) 2 = 1 s 4 + 2 s 2 t 2 + t 4 = 1 p ( s 2 + t 2 ) 2 = 1 s 2 + t 2 . Now by the change of variables formula ( § 15.10): ZZ D dxdy p x 2 + y 2 = ZZ { s 2 + t 2 1 ,s 0 ,t 0 } 4( s 2 + t 2 ) s 2 + t 2 dA = 4 ZZ { s 2 + t 2 1 ,s 0 ,t 0 } dA = 4 ² π 4 ³ = π since the area of the quater circle in the st -plane is π 4 . 10. Let F = z 2 i + ze yz j + (2 xz + cos z + ye yz ) k . Find a function f ( x,y,z ) so that f = F .
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This note was uploaded on 11/16/2011 for the course MULTIVARIA 251 taught by Professor Staff during the Fall '11 term at Rutgers.

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practice32answers - Practice Exam 2 Solutions The answer to...

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