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20EFinalSolns!!!

# 20EFinalSolns!!! - 1(6 points Evaluate 1 1 11 y=o sin(x 3...

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1. (6 points) Evaluate 1 1 11 sin (x 3 ) dxdy. y=o x=..fji by changing the order of integration. (Note: you need not simplify trigonometric expressions in your final answer.) , I I ' I XL , t SV'""'i Sinlx:'')dxd\ -:: So So S\l'Ilx.~) c1~dx: j ~ i CJ 2 Y -"'IV) ~ ~ :-f) .c--J. -2) if J' \J, (j) s~ >l &X ~ ~ I J ~cJ ~ ""5 -- Q ~ ~~,~ ~:t E fi ~ .>< 57 ~ j Q - e ~ £ ~ --Q <:J \$ =~J~H l x~ -r S~x3, I 0 d x \ vt=x'3 Xl cs ir) ",3 d;>< d~ = 3,,2dx ~: X =0 -=t 4 ::.6 l X -=., =::, \A :::.. J - - J() r ~ Vrtu d~ _\ "3 LDS u. \ 0 1 - \ , -- eD~ \ ~- Cos () 3 3 -\ 3 (oS, \- l. 3

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2. (6 points) Find an appropriate change of variables to evaluate J In (x + y) eZ'-i' dxdy, where R is the square with vertices (2,0), (0,2), (-2,0), (0, -2). (Hint: first determine equations for each side of the square R.) /'1-- v :' ',. ~ ~ ~ ~-:; u+ --- v --2<\1..<"''- "" = X Jr "-'\ 2- \j -= 'X- 'j ~ LA-V -L,zV<'2 ~:= 1- I J )(/d~ dx/dv \/2 til.' _ t \ -= - \ t:t-~ 2.. 1 d~/dv. ~~" \/2. -:"2 Sf ~2 '12 d R. Lx \-~l e Xd~ -= 5S u e U-v (-i~ 'd V d LA -::. 121~ \Ae ~" ci" d'" -:: 12 \) ~elA~ \:2- cl~ -') lA -2 -2 -2- 2. \.4 -24 \ 1- - '2,-,\ _ ~'2. 2u e __ E:, - - e -e du - -2 . 2- -2- '?I..\ 2,,\ \ 1- _ s- \-e - - e2~\: -= le.Lt y _ - ~ - 2- -2 I'r -, /~ '), I'- 'Ix ~ &~,~ -r+~_ '< 1- -2.. ~-:;. y..-7- )<_'-\~ J, e-l1~
3. Consider the vector field F(x, y, z) (x - yzsin(xyz), y - xzsin(xyz), -xysin(xyz)). (a) (3 points) Find a scalar function /(x, y, z) such that F V /.

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