exam2sample1 - MATH 32B SECOND MIDTERM LAST NAME FIRST NAME...

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MATH 32B SECOND MIDTERM MAY 26, 2004 LAST NAME FIRST NAME ID NO. Please write clearly and legibly. There are four 20-point problems and four multiple choice problems worth 5 points each. To receive credit, you must circle your answer . DO NOT WRITE BELOW THIS LINE 1 5 2 6 3 7 4 8 TOTAL
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2 PROBLEM 1 (20 Points) Use the mapping T ( u, v ) = ( u - 2 v, v ) and the change of variables formula to compute integraldisplay integraldisplay R ( x + 4 y ) dx dy where R is the shaded region in the Figure. CIRCLE YOUR ANSWER x+2y = 10 6 x y 10 3 5 1 x+2y = 6 Solution: We have x = u - 2 v and y = v , so u = x + 2 y . The region R is defined by the inequalities 6 u 10 , 1 v 3 Furthermore, vextendsingle vextendsingle vextendsingle ( x, y ) ( u, v ) vextendsingle vextendsingle vextendsingle = vextendsingle vextendsingle vextendsingle vextendsingle 1 - 2 0 1 vextendsingle vextendsingle vextendsingle vextendsingle = 1 so the Jacobian factor is 1. Since x + 4 y = u + 2 v , the change of variables formula gives us: integraldisplay integraldisplay R ( x + 3 y ) dx dy = integraldisplay 10 u =6 integraldisplay 3 v =1 ( u + 2 v ) dudv = 1 2 (10 2 - 6 2 )(2) + 4(3 2 - 1 2 ) = 64 + 32 = 96
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3 PROBLEM 2 (20 Points) Compute
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