2009 Exam2 Solution

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

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Unformatted text preview: 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 ZZ R xy ( y 2 + 1) 2 dA. SOLUTION: The iterated integral is R 1 R 3 xy ( y 2 +1) 2 dxdy. The inside integral is R 3 xdx = 9 2 , so the double integral is 9 2 Z 1 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 Z 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 RR R f ( x,y ) dA ? SOLUTION: Evaluate ZZ R xy 2 dA = Z 2 Z 3 xy 2 dxdy = [ x 2 / 2] 3 x =0 [ y 3 / 3] 2 y =0 = 9 2 8 3 = 12 . Also, ZZ R (- 1) dA =- A ( R ) =- 6 , so- 6 ≤ ZZ 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 ) ≡ if x 2 + y 2 > 1 . (a) (10 points) What does the constant α need to be? (Your answer will involve ln2. Do not evaluate ln2.) SOLUTION: Compute in polar coordinates, and substitute...
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This note was uploaded on 10/02/2011 for the course MATH 2263 taught by Professor Staff during the Spring '08 term at Minnesota.

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

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