2220pracprelim2 - Math 2220 Practice Prelim II Spring 2010...

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Unformatted text preview: Math 2220 Practice Prelim II Spring 2010 Calculators, books, and notes are not permitted. Show all your work. Formulas that might be useful: cos2 θ = 1 (1 + cos 2θ ) 2 sin2 θ = 1 (1 − cos 2θ ) 2 1. Sketch the region of integration for the integral 2 −2 4 y2 0 4−x dz dx dy and set up the integral in the order dy dz dx . (Include all the limits of integration.) Do not evaluate the integral. 2. Evaluate the integral e−x R 2 −y 2 −z 2 x2 + y 2 dx dy dz where R is the ball x2 + y 2 + z 2 ≤ a2 of radius a (an arbitrary constant). 3. Using polar coordinates, find the area of the region inside the circle (x − 1)2 + y 2 = 1 and outside the circle x2 + y 2 = 1 . 4. Sketch the region in R3 defined by the inequalities y 2 + z 2 ≤ 1 , x ≤ y , x ≥ 0 , and z ≥ 0 and compute the volume of this region. 5. Use an appropriate change of variables to evaluate 1 R 4 x2 + 9 y 2 dA where R is the region inside the ellipse 4x2 + 9y 2 = 1 and above the line 3y = −2x . 6. Find the center of mass of the upper hemisphere x2 + y 2 + z 2 ≤ a2 , z ≥ 0 , with constant density δ = 1 . ...
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