MTH 162 Final: MTH 162 University of Rochester Final Solutions Spring2007
31 Jan 2019
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Part i: (10 points) (a) find the area enclosed by the curves y = x + 2 and y = x2. (b) find the volume of the solid obtained by rotating this same region about the x-axis. Answer: (a) first we need to nd where these curves intersect. = (2 + 4 8/3) (1/2 2 + 1/3) = 9/2 (b) when we rotate about the x-axis, a typical vertical cross section is a washer with outer radius x + 2 and inner radius x2. Thus the area is given by a(x) = [(x + 2)2 (x2)2] = 1(x2 + 4x + 4 x4) dx. = (x3/3 + 2x2 + 4x x5/5)|2. = [(8/3 + 8 + 8 32/5) ( 1/3 + 2 4 + 1/5)] = 72 /5: (10 points) gasoline at a service station is stored in a cylindrical tank buried on its side, with the highest part of the tank 5 ft below the surface.
