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Disc4 - Math 21B Kouba Discussion Sheet 4 1 Consider region...

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Unformatted text preview: Math 21B Kouba Discussion Sheet 4 1.) Consider region R bounded by the graphs of y = 3x , y = a: , and a: = 1. a.) Compute the area of R. b.) Use the DISC METHOD (Set up only.) to find the volume of the solid formed by revolving R about the i.) x—axis. ii.) y-axis. iii.) line y = 4. iv.) line x = -1. 0.) Repeat part b.) using the SHELL METHOD. 2.) Consider region R bounded by the graphs of a: = y2 and a: = y3 — yz. a.) Find the area of R. b.) Use the Disc Method or the Shell Method (Pick the more convenient method and set up only.) to find the volume of the solid formed by revolving R about the i.) x—axis. ii.) line x = -2. y 3.) Consider the region R given in the diagram at right. d If the volume of the solid formed by revolving R about the y-axis is 101r and the volume of the solid formed by revolving R about the line x = -1 is 20w, what is the area of R ? 4.) Compute the arc length of the given curve on the indicated interval. a.) y 2 (ES/4 on the interval [0,1] b.) a: 2 cost + tsint and y = sint — tcost on the interval [7r/6, 7r/4] 5.) A thin rod lies on the z-axis from a: = 0 to :1: = 20 centimeters. Assume that the density of the rod 3; centimeters from its left end is 6(23) 2 Va: + 1 grams per centimeter. a.) Determine the rod’s total mass. b.) Determine the rod’s center of mass. 6.) A thin plate lies in the region bounded by the graphs of y = $2, x = 2, and y = 0. Assume that the density at point (x, y) of the plate is 6(23, y) = 10 ounces per square inch. a.) Determine the plate’s total mass. b.) Determine the plate’s center of mass. 7.) A thin plate lies in the region bounded by the graphs of y = 6"”, y = 1, and a: = 1n 2. Assume that the density at point (:12, y) of the plate is 6(53, y) = $2 + 1 ounces per square inch. a.) Determine the plate’s total mass. b.) Determine the plate’s center of mass. 4 5 8.) Find the centroid of the region bounded by the graphs of y = z and y = x . THE FOLLOWING PROBLEM IS FOR RECREATIONAL PURPOSES ONLY. 9.) Determine the exact value of the following expression : 1 + 1 + 1 + 1 + \/1 + ...
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