32B(W08)_Midterm1_Solutions

32B(W08)_Midterm1_Solutions - MATH 32B - Lecture 4 - Winter...

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MATH 32B - Lecture 4 - Winter 2008 Solutions to Midterm 1 - February 4, 2008 NAME: STUDENT ID #: DISCUSSION SECTION: 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. POINTS: 1. 2. 3. 4. 5. 6. 1
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2 1.(15 points) Evaluate ZZZ E yz dV where E lies above the plane z = 0, below the plane z = y , and inside the cylinder x 2 + y 2 = 4. Solution. = Z 2 - 2 Z 4 - x 2 0 Z y 0 yz dzdydx = Z 2 - 2 Z 4 - x 2 0 y 3 2 dydx = Z 2 - 2 2 - x 2 + x 4 8 dx = 64 15 /
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3 2.(20 points) Find the area of the part of the paraboloid z = 4 - x 2 - y 2 that lies within the cylinder x 2 + y 2 = 1 Solution. A ( S ) = ZZ D q 1 + f 2 x + f 2 y dA z = f ( x,y ) = 4 - x 2 - y 2 f x = - 2 x,f y = - 2 y A ( S ) = ZZ D p 1 + 4 x 2 + 4 y 2 dA = Z 2 π 0 Z 1 0 p 1 + 4 r 2 r drdθ = Z 2 π 0 Z 1 0 r p 1 + 4 r 2 dr = (2 π )( 5 5 - 1 12 ) = π (5 5 - 1) 6 /
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This note was uploaded on 05/01/2008 for the course MATH 32B taught by Professor Rogawski during the Winter '08 term at UCLA.

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32B(W08)_Midterm1_Solutions - MATH 32B - Lecture 4 - Winter...

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