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Unformatted text preview: choi (kc24547) – HW05 – BERG – (56525) 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 = y about the y-axis between y = 0 and y = 3. 1. V = 7 π cu.units 2. V = 5 π cu.units 3. V = 9 π cu.units correct 4. V = 8 π cu.units 5. V = 6 π 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 = π integraldisplay b a f ( y ) 2 dy. When f ( y ) = y and a = 0 , b = 3, therefore, V = π integraldisplay b a y 2 dx = π bracketleftBig 1 3 y 3 bracketrightBig 3 . Consequently, V = 9 π 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 = 2 about the x-axis. 1. volume = 31 π cu.units 2. volume = 30 π cu.units 3. volume = 32 π cu.units correct 4. volume = 29 π cu.units 5. volume = 33 π 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 = π integraldisplay b a f ( x ) 2 dx . When f ( x ) = 4 √ x, a = 0 , b = 2 , therefore, V = π integraldisplay 2 16 x dx = π 2 bracketleftBig 16 x 2 bracketrightBig 2 . Consequently, V = 32 π 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 y-axis. (Hint: as always graph the region first ). 1. V = 27 cu. units 2. V = 81 2 cu. units 3. V = 27 π cu. units 4. V = 81 2 π cu. units correct choi (kc24547) – HW05 – BERG – (56525) 2 5. V = 81 4 cu. units 6. V = 81 4 π cu. units Explanation: The region rotated about the y-axis 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 y-axis about the y-axis is given by volume = π 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 = π integraldisplay 9 y dy = π bracketleftbigg 1 2 y 2 bracketrightbigg 9 . Consequently, V = 81 2 π . 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 + 5 = 0 . 1. volume = 22 3 π 2. volume = 16 3 π 3. volume = 10 3 π correct 4. volume = 13 3 π 5. volume = 19 3 π Explanation: The solid is obtained by rotating the shaded region about the line x + 5 = 0 as shown in 1 x + 5 = 0 (not drawn to scale). To compute the volume of this solid we use the washer method. For this we have to express f and g as functions...
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- Spring '10