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CalcReview2 - Version 010 Review 2 Odell(58340 This...

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Version 010 – Review 2 – Odell – (58340) 1 This print-out should have 16 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Evaluate the integral I = integraldisplay 3 1 3 15 + 2 x - x 2 dx . 1. I = 3 4 π 2. I = 1 2 3 3. I = π 4. I = 3 5. I = 1 2 π correct 6. I = 3 4 3 Explanation: By completing the square we see that 15 + 2 x - x 2 = 16 - ( x - 1) 2 , so I = integraldisplay 3 1 3 radicalbig 16 - ( x - 1) 2 , dx . Now set x - 1 = 4 sin u . Then dx = 4 cos u du , while x = 1 = u = 0 , x = 3 = u = π 6 . Thus I = 3 integraldisplay π/ 6 0 4 cos u 4 cos u du = bracketleftBig 3 u bracketrightBig π/ 6 0 . Consequently, I = 1 2 π . 002 10.0 points Evaluate the definite integral I = integraldisplay 3 0 sin 1 x 3 dx . 1. I = 3 4 ( π - 2 ln 2) 2. I = 3 3. I = 3 2 ( π - 2) correct 4. I = - 3 2 5. I = 3 2 ( π - 1) 6. I = 3 4 ( π + 2 ln 2) Explanation: Let x = 3 u ; then dx = 3 du while x = 0 = u = 0 , x = 3 = u = 1 . In this case, I = 3 integraldisplay 1 0 sin 1 u du , so after integration by parts, I = 3 bracketleftBig u sin 1 u bracketrightBig 1 0 - 3 integraldisplay 1 0 u 1 - u 2 du = 3 bracketleftBig u sin 1 u + ( 1 - u 2 ) 1 / 2 bracketrightBig 1 0 . Consequently, I = 3 parenleftBig π 2 - 1 parenrightBig = 3 2 ( π - 2) . 003 10.0 points
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Version 010 – Review 2 – Odell – (58340) 2 Reverse the order of integration in the inte- gral I = integraldisplay 3 1 parenleftBig integraldisplay 1 x 2 / 3 f ( x, y ) dy parenrightBig dx, but make no attempt to evaluate either inte- gral. 1. I = integraldisplay 1 1 3 parenleftBig integraldisplay 1 3 y f ( x, y ) dx parenrightBig dy 2. I = integraldisplay 1 1 3 parenleftBig integraldisplay 3 y 1 f ( x, y ) dx parenrightBig dy correct 3. I = integraldisplay 1 y 2 / 3 parenleftBig integraldisplay 3 1 f ( x, y ) dx parenrightBig dy 4. I = integraldisplay 3 1 parenleftBig integraldisplay y 3 f ( x, y ) dx parenrightBig dy 5. I = integraldisplay 1 1 3 parenleftBig integraldisplay 3 y f ( x, y ) dx parenrightBig dy Explanation: The region of integration is similar to the shaded region in the figure y x (not drawn to scale). This shaded region is enclosed by the graphs of 3 y = x 2 , y = 1 , x = 1 . In the given order of integration, first x is fixed and then y varies along the solid line from y = x 2 / 3 to y = 1. To change the order of integration, first fix y . Then, x varies along the dashed line from x = 1 to x = radicalbig 3 y . To cover the region of integration, therefore, y must now vary from 1 3 to 1. Hence, after changing the order of integration, I = integraldisplay 1 1 3 parenleftBig integraldisplay 3 y 1 f ( x, y ) dx parenrightBig dy . keywords: 004 10.0 points Find the value of the definite integral I = integraldisplay π/ 4 0 parenleftBig 4 sec 4 x - 3 sec 2 x parenrightBig tan x dx .
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