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Unformatted text preview: Miller, Kierste – Homework 4 – Due: Feb 13 2007, 3:00 am – Inst: Gary Berg 1 This printout should have 21 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Find the volume of the paraboloid gener ated by rotating the parabola y = 8 √ x about the xaxis between x = 0 and x = 1. 1. V = 32 π cu.units correct 2. V = 36 π cu.units 3. V = 34 π cu.units 4. V = 35 π cu.units 5. V = 33 π cu.units Explanation: The volume, V , of the solid of revolution generated by rotating the graph of y = f ( x ) about the xaxis between x = a and x = b is given by V = π Z b a f ( x ) 2 dx. When f ( x ) = 8 √ x and a = 0 , b = 1, there fore, V = π Z 1 64 xdx = π 2 h 64 x 2 i 1 . Conequently, V = 32 π cu.units . keywords: volume, integral, solid of revolu tion 002 (part 1 of 1) 10 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 = 27 cu. units 2. V = 27 π cu. units 3. V = 81 2 cu. units 4. V = 81 4 cu. units 5. V = 81 4 π cu. units 6. V = 81 2 π cu. units correct 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 = π Z 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 Z 9 y dy = π • 1 2 y 2 ‚ 9 . Miller, Kierste – Homework 4 – Due: Feb 13 2007, 3:00 am – Inst: Gary Berg 2 Consequently, V = 81 2 π . keywords: volume, integral, solid of revolu tion 003 (part 1 of 1) 10 points Find the volume, V , of the solid obtained by rotating the bounded region in the first quadrant enclosed by the graphs of y = x 3 2 , x = y 3 about the xaxis. 1. V = 13 30 π cu. units 2. V = 2 5 π cu. units 3. V = 13 30 cu. units 4. V = 2 5 cu. units 5. V = 7 20 π cu. units correct 6. V = 7 20 cu. units Explanation: Since the graphs of y = x 3 2 , x = y 3 intersect at (0 , 0) and at (1 , 1) the bounded region in the first quadrant enclosed by their graphs is the shaded area shown in 1 1 Thus the volume of the solid of revolution generated by rotating this region about the xaxis is given by V = π Z 1 n ( x 1 / 3 ) 2 ( x 3 2 ) 2 o dx = π Z 1 n x 2 3 x 3 o dx = π • 3 5 x 5 3 1 4 x 4 ‚ 1 . Consequently, V = π ‡ 3 5 1 4 · = 7 20 π cu. units ....
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 Spring '08
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