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32B(W08)_PracticeMidterm1_Solutions

# 32B(W08)_PracticeMidterm1_Solutions - MATH 32B Lecture 4...

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MATH 32B - Lecture 4 - Winter 2008 Solutions to Practice Midterm 1 - January 30, 2008 NAME: STUDENT ID #: This is a closed-book and closed-note examination. Calculators are not allowed. Please show all your work. Use only the paper provided. You may write on the back if you need more space, but clearly indicate this on the front. There are 6 problems for a total of 100 points. 1

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2 1.(15 points) Evaluate Z Z Z E x 2 e y dV where E is bounded by the parabolic cylinder z = 1 - y 2 and the planes z = 0, x = 1 and x = - 1. Solution. Z 1 - 1 Z - 1 - 1 Z 1 - y 2 0 x 2 e y dzdydx = Z 1 - 1 Z - 1 - 1 (1 - y 2 ) x 2 e y dydx = 8 e - 1 3 /
3 2.(20 points) Find the area of the part of the sphere x 2 + y 2 + z 2 = a 2 that lies within the cylinder x 2 + y 2 = ax and above the xy -plane. Solution. z = f ( x, y ) = p a 2 - x 2 - y 2 A ( S ) = Z Z D q 1 + f 2 x + f 2 y dA f x = - x p a 2 - x 2 - y 2 f y = - y p a 2 - x 2 - y 2 A ( S ) = Z π 2 - π 2 Z a cos( θ ) 0 r 1 + r 2 a 2 - r 2 rdrdθ = Z π 2 - π 2 Z ( a 2 ) sin 2 ( θ ) a 2 - a 2 u dudθ = a 2 ( π - 2) /

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4 3.(15 points) Calculate the iterated integral Z 1 0 Z 1 y cos(3 x 2 ) dxdy Solution.
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