practice_final_fa05_solutions - Practice Final Exam, Math...

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Unformatted text preview: Practice Final Exam, Math 191, Fall 2005 No calculators. Show your working. Clearly mark each answer. 1. (a) Method I. The arc length is integraldisplay 2 radicalBigg 1 + parenleftbigg dy dx parenrightbigg 2 dx = integraldisplay 2 radicalBigg 1 + parenleftbigg 2 − 2 x 2 √ 2 x − x 2 parenrightbigg 2 dx = integraldisplay 2 radicalbigg 1 + (1 − x ) 2 2 x − x 2 dx = integraldisplay 2 radicalbigg 1 2 x − x 2 dx = integraldisplay 2 1 radicalbig 1 − ( x − 1) 2 dx = integraldisplay 1 − 1 1 √ 1 − u 2 du = bracketleftbig sin − 1 u bracketrightbig 1 − 1 = π 2 − parenleftBig − π 2 parenrightBig = π Method II. y = √ 2 x − x 2 = radicalbig 1 − ( x − 1) 2 , ≤ x ≤ 2 traces out the half of the radius 1 circle centred at (1 , 0) , that lies above the x-axis, and this has length π . (b) See page 536 of the textbook for the sketch. The volume is integraldisplay ∞ −∞ π y 2 dx = integraldisplay ∞ −∞ π sech 2 x dx = lim b → + ∞ integraldisplay ∞ π sech 2 x dx + lim a →−∞ integraldisplay a π sech 2 x dx = lim b → + ∞ [ π tanh x ] b + lim a →−∞ [ π tanh x ] a = π lim b → + ∞ e b − e − b e b + e − b − 0 + 0 − π lim a →−∞ e a − e − a e a + e − a = π lim b → + ∞ 1 − e − 2 b 1 + e − 2 b − 0 + 0 − π lim a →−∞ e 2 a − 1 e 2 a + 1 = π. 1 − π. ( − 1) = 2 π (c) The surface area is integraldisplay 2 2 πy radicalBigg 1 + parenleftbigg dy dx parenrightbigg 2 dx = 2 π integraldisplay 2 cosh x radicalbig 1 + sinh 2 x dx = 2 π integraldisplay 2 cosh 2 x dx = 2 π integraldisplay 2 parenleftbigg e x + e − x 2 parenrightbigg 2 dx = 2 π integraldisplay 2 e 2 x + 2 + e − 2 x 4 dx = 2 π bracketleftbigg e 2 x + 4 x − e − 2 x 8 bracketrightbigg 2 = 2 π e 4 + 8 − e − 4 8 = π e 4 + 8 − e − 4 4 (Alternatively, use the identity 2 cosh 2 x = 1 + cosh 2 x in this calculation.) 2.2....
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This test prep was uploaded on 02/15/2008 for the course MATH 1910 taught by Professor Berman during the Fall '07 term at Cornell University (Engineering School).

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practice_final_fa05_solutions - Practice Final Exam, Math...

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