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Unformatted text preview: gutierrez (ig3472) HW05 Neitzke (56585) 1 This print-out should have 22 questions. Multiple-choice 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 = 3 4 y about the y-axis between y = 0 and y = 4. 1. V = 10 cu.units 2. V = 13 cu.units 3. V = 12 cu.units correct 4. V = 9 cu.units 5. V = 11 cu.units Explanation: The volume, V , of the solid of revolution generated by rotating the graph of x = f ( y ) about the y-axis between y = a and y = b is given by V = integraldisplay b a f ( y ) 2 dy. When f ( y ) = 3 4 y and a = 0 , b = 4, therefore, V = integraldisplay b a 9 16 y 2 dx = bracketleftBig 3 16 y 3 bracketrightBig 4 . Consequently, V = 12 cu.units . 002 10.0 points Find the volume of the paraboloid gener- ated by rotating the graph of y = 6 x be- tween x = 0 and x = 2 about the x-axis. 1. volume = 70 cu.units 2. volume = 71 cu.units 3. volume = 68 cu.units 4. volume = 72 cu.units correct 5. volume = 69 cu.units Explanation: The solid of revolution generated by rotat- ing the graph of y = f ( x ) about the x-axis between x = a and x = b has volume = integraldisplay b a f ( x ) 2 dx . When f ( x ) = 6 x, a = 0 , b = 2 , therefore, V = integraldisplay 2 36 x dx = 2 bracketleftBig 36 x 2 bracketrightBig 2 . Consequently, V = 72 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 = 4 about the y-axis. (Hint: as always graph the region first ). 1. V = 4 cu. units 2. V = 16 3 cu. units 3. V = 16 3 cu. units 4. V = 8 cu. units gutierrez (ig3472) HW05 Neitzke (56585) 2 5. V = 4 cu. units 6. V = 8 cu. units correct Explanation: The region rotated about the y-axis is sim- ilar to the shaded region in 4 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 y-axis about the y-axis 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 4 y dy = bracketleftbigg 1 2 y 2 bracketrightbigg 4 . Consequently, V = 8 . 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 + 4 = 0 . 1. volume = 11 15 2. volume = 71 15 3. volume = 41 15 correct 4. volume = 26 15 5. volume = 56 15 Explanation: The solid is obtained by rotating the shaded region about the line x + 4 = 0 as shown in 1 x + 4 = 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 04/13/2010 for the course MATH 408L taught by Professor Gogolev during the Spring '09 term at University of Texas at Austin.
- Spring '09