mt166f2009_2 - 2(0 0 and(2 0 Find the centroid of the plate...

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Math 166 Midterm Exam Part II October 8, 2009 Full Name: Instructor & Section: Instructions: Answer each question completely. Show all work. No credit is allowed for mere answers with no work shown. Show the steps of calculations. State the reasons that justify conclusions. Give exact values in results. 1. (16 points) A region R is bounded by the graph of y = e x , the line y = 1, and the line x = 3. Sketch the region R . (a) A solid is generated by revolving the region R about the line x = - 1. Set up, but do not evaluate, a definite integral for the volume of the resulting solid of revolution. (b) A second solid has base R , and cross-sections perpendicular to the x -axis are disks with a diameter in R . Set up, but do not evaluate, a definite integral for the volume of this region.
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Math 166 Midterm Part II Page 2 2. (16 points) A thin metal plate in the plane occupies the triangle with vertices (0
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Unformatted text preview: , 2), (0 , 0), and (2 , 0). Find the centroid of the plate. You may use symmetry and known area or volume formulas from geometry to evaluate integrals. Math 166 Midterm Part II Page 3 3. (16 points) An upright water tank has a flat base in the plane and is filled with water. The tank has height 7 feet, and, for 0 ≤ x ≤ 7, the cross-section parallel to the base at height x has area x 2 + 2. How much work is done in pumping all the water over the top edge of the tank? [Use δ lb / ft 3 for the weight-density of water.] Math 166 Midterm Part II Page 4 4. (16 points) Let G be the portion of the graph with parametric equations x = 1 + sin t, y = 3 + 2 cos t, for 0 ≤ t ≤ 2 π . Set up, but do not evaluate, an integral for (a) the length of G . (b) the area of the surface generated by revolving G about the x-axis....
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