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Unformatted text preview: Perez (nap563) HW05 Zheng (56555) 1 This printout should have 22 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 10.0 points Determine the volume of the right circular cone generated by rotating the line x = 4 3 y about the yaxis between y = 0 and y = 3. 1. V = 17 cu.units 2. V = 15 cu.units 3. V = 16 cu.units correct 4. V = 14 cu.units 5. V = 13 cu.units Explanation: The volume, V , of the solid of revolution generated by rotating the graph of x = f ( y ) about the yaxis between y = a and y = b is given by V = integraldisplay b a f ( y ) 2 dy. When f ( y ) = 4 3 y and a = 0 , b = 3, therefore, V = integraldisplay b a 16 9 y 2 dx = bracketleftBig 16 27 y 3 bracketrightBig 3 . Consequently, V = 16 cu.units . 002 10.0 points Find the volume of the paraboloid gener ated by rotating the graph of y = 4 x be tween x = 0 and x = 1 about the xaxis. 1. volume = 11 cu.units 2. volume = 10 cu.units 3. volume = 8 cu.units correct 4. volume = 7 cu.units 5. volume = 9 cu.units Explanation: The solid of revolution generated by rotat ing the graph of y = f ( x ) about the xaxis between x = a and x = b has volume = integraldisplay b a f ( x ) 2 dx . When f ( x ) = 4 x, a = 0 , b = 1 , therefore, V = integraldisplay 1 16 x dx = 2 bracketleftBig 16 x 2 bracketrightBig 1 . Consequently, V = 8 cu.units . keywords: volume, integral, solid of revolu tion 003 10.0 points Find the volume, V , of the solid obtained by rotating the region bounded by y = x 2 , x = 0 , y = 9 about the yaxis. (Hint: as always graph the region first ). 1. V = 81 2 cu. units 2. V = 81 4 cu. units 3. V = 81 4 cu. units 4. V = 81 2 cu. units correct Perez (nap563) HW05 Zheng (56555) 2 5. V = 27 cu. units 6. V = 27 cu. units Explanation: The region rotated about the yaxis is sim ilar to the shaded region in 9 y x (not drawn to scale). Now the volume of the solid of revolution generated by revolving the graph of x = f ( y ) on the interval [ a, b ] on the yaxis about the yaxis is given by volume = integraldisplay b a f ( y ) 2 dy . To apply this we have first to express x as a function of y since initially y is defined in terms of x by y = x 2 . But after taking square roots we see that x = y 1 / 2 . Thus V = integraldisplay 9 y dy = bracketleftbigg 1 2 y 2 bracketrightbigg 9 . Consequently, V = 81 2 . 004 10.0 points Let A be the bounded region enclosed by the graphs of f ( x ) = x , g ( x ) = x 4 . Find the volume of the solid obtained by ro tating the region A about the line x + 2 = 0 . 1. volume = 22 15 2. volume = 83 15 3. volume = 68 15 4. volume = 23 15 correct 5. volume = 37 15 Explanation: The solid is obtained by rotating the shaded region about the line x + 2 = 0 as shown in 1 x + 2 = 0 (not drawn to scale). To compute the volume of this solid we use the washer method. For this we have to express...
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This note was uploaded on 03/17/2010 for the course MATH 408 L taught by Professor Zheng during the Spring '10 term at University of Texas at Austin.
 Spring '10
 ZHENG
 Math, Calculus

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