2009 Exam2 Solution

2009 Exam2 Solution - Math 2263 Fall 2009 Midterm 2 WITH...

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Math 2263 Fall 2009 Midterm 2, WITH SOLUTIONS November 5, 2009 1. (7 points) Let R be the rectangle 0 x 3, 0 y 1. Find the double integral R xy ( y 2 + 1) 2 dA. SOLUTION: The iterated integral is 1 0 3 0 xy ( y 2 +1) 2 dx dy. The inside integral is 3 0 x dx = 9 2 , so the double integral is 9 2 1 0 y ( y 2 + 1) 2 dy. Make the substitution u = y 2 + 1, so du = 2 y dy and the answer is 9 2 1 2 2 1 du u 2 = 9 4 [ - u - 1 ] 2 1 = 9 8 . 2. (8 points) Let R be the rectangle 0 x 5, 0 y 2 in the ( x, y )-plane. If a continuous function f ( x, y ) satisfies - 1 f ( x, y ) xy 2 , what does this tell you about the value of R f ( x, y ) dA ? SOLUTION: Evaluate R xy 2 dA = 2 0 3 0 xy 2 dx dy = [ x 2 / 2] 3 x =0 [ y 3 / 3] 2 y =0 = 9 2 8 3 = 12 . Also, R ( - 1) dA = - A ( R ) = - 6 , so - 6 R f ( x, y ) dA 12 . 3. (20 points) Suppose x = X and y = Y are random variables with joint density function f ( x, y ) = α 1 + x 2 + y 2 if x 2 + y 2 1 , and f ( x, y ) 0 if x 2 + y 2 > 1 . (a) (10 points) What does the constant α need to be? (Your answer will involve ln 2. Do not evaluate ln 2.) SOLUTION: Compute in polar coordinates, and substitute u = 1 + r 2 : f ( x, y ) dA = 2 π 0 1 0 α 1+ r
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