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080228quiz5solution

080228quiz5solution - R(a about the line y = 6 If we use...

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MATH 126 QUIZ #5 Solution 02/28/08 Mathias Knape Last Name : Circle discussion section First Name: 11 am 12 pm 1 pm Note: In order to receive full credit, please show all your work (conceptual steps, definitions you use, ...) 1. Find the length of the curve y = cosh x , 0 x 1. The length of the curve is given by L = 1 0 1 + dy dx 2 dx = 1 0 1 + sinh 2 xdx = 1 0 cosh 2 xdx = 1 0 cosh xdx = sinh x | 1 0 = sinh 1 sinh 0 = sinh 1 2. Use the method of cylindrical shells to find the volume generated by rotating the region bounded by y = x 2 , y = 0, x = 2 and x = 1 about the y-axis. Sketch the region and a typical shell. V = - 1 - 2 2 π ( x ) x 2 dx = 2 π - 1 - 2 x 3 dx = 2 π [ 1 4 x 4 | - 1 - 2 = 2 π ( 1 4 + 4) = 15 2 π
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3. Consider the region R bounded by y = x 2 6 and y = x . Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region
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Unformatted text preview: R (a) about the line y = 6. If we use the washer method we obtain V = π Z 3-2 (12 − x 2 ) 2 − (6 − x ) 2 dx If we use the cylindrical shell method we get V = Z-2-6 2 π (6 − y )2 p y + 6 dy + Z 3-2 2 π (6 − y ) ± p y + 6 − y ² dy (b) about the line x = − 6. We can use both methods again, but the cylindrical shell method is in this case more convenient. V = Z 3-2 2 π ( x + 6)( x − ( x 2 − 6)) dx In case you used the washer method you should have V = Z-2-6 π ³ ± 6 + p y + 6 ² 2 − ± 6 − p y + 6 ² 2 ´ dy + Z 3-2 π ³ ± 6 + p y + 6 ² 2 − (6 + y ) 2 ´ dy...
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