Practice exam3 Solution

Practice exam3 Solution - Math 2263 Spring 2009 Midterm 3...

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Unformatted text preview: Math 2263 Spring 2009 Midterm 3, WITH SOLUTIONS April 23, 2009 1. (10 points) Let B be the box, or rectangular solid: 0 ≤ x ≤ 2, 0 ≤ y ≤ 3, 0 ≤ z ≤ 1. Find ZZZ B ( xy- 2 yz + x 2 z 2 ) dV. SOLUTION: ZZZ B ( xy- 2 yz + x 2 z 2 ) dV = Z 1 Z 3 h x 2 y 2- 2 yzx- x 3 z 2 3 i 2 x =0 dy dz = Z 1 Z 3 2 y- 4 yz + 8 z 2 3 dy dz = Z 1 h y 2- 2 y 2 z + 8 z 2 y 3 i 3 y =0 dz = Z 1 (9- 18 z +8 z 2 ) dz = 9- 9+ 8 3 = 8 3 . 2. (15 points) Let E be the solid region bounded below by the cone z 2 = x 2 + y 2 and above by the plane z = 1. Find ZZZ E ( x 2 + y 2 ) dV. Hint: Try cylindrical coordinates. SOLUTION: Use cylindrical coordinates: E is described by 0 ≤ r ≤ z ≤ 1, 0 ≤ θ ≤ 2 π . So ZZZ E ( x 2 + y 2 ) dV = Z 1 Z 2 π Z z r 2 r dr dθ dz = 2 π Z 1 z 4 4 dz = π 10 . 3. (25 points) C 1 and C 2 are oriented curves in the ( x,y )-plane, each of which starts at (0 , 0) and ends at (1 , 1). C 1 is given by y = x 2 , 0 ≤ x ≤ 1; and C 2 is given by x = y 2 , 0 ≤ y ≤ 1. Let the1....
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Practice exam3 Solution - Math 2263 Spring 2009 Midterm 3...

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