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Unformatted text preview: moseley (cmm3869) HW16 Gilbert (56380) 1 This printout should have 16 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 10.0 points By changing to polar coordinates evaluate the integral I = integraldisplay integraldisplay R radicalbig x 2 + y 2 dxdy when R is the region braceleftBig ( x, y ) : 9 x 2 + y 2 25 , y bracerightBig in the xyplane. 1. I = 86 3 2. I = 95 3 3. I = 92 3 4. I = 89 3 5. I = 98 3 correct Explanation: In polar cooordinates, R = braceleftBig ( r, ) : 3 r 5 , bracerightBig , while I = integraldisplay integraldisplay R r ( rdrd ) = integraldisplay integraldisplay R r 2 drd , since radicalbig x 2 + y 2 = r . But then I = integraldisplay 5 3 integraldisplay r 2 drd = integraldisplay 5 3 r 2 dr . Consequently, I = 1 3 bracketleftBig r 3 bracketrightBig 5 3 = 98 3 . 002 10.0 points By changing to polar coordinates evaluate the integral I = integraldisplay integraldisplay R 4 e x 2 y 2 dxdy when R is the region in the xyplane bounded by the graph of x = radicalbig 4 y 2 and the yaxis. 1. I = 2 (1 e 4 ) correct 2. I = 4 (1 e 4 ) 3. I = 2 (1 e 2 ) 4. I = (1 e 4 ) 5. I = (1 e 2 ) 6. I = 4 (1 e 2 ) Explanation: In polar cooordinates, R is the set braceleftBig ( r, ) : 0 r 2 , 2 2 bracerightBig , while I = integraldisplay integraldisplay R 4 e r 2 ( rdrd ) = integraldisplay integraldisplay R 4 re r 2 drd , since x 2 + y 2 = r 2 . But then I = 4 integraldisplay 2 integraldisplay / 2 / 2 re r 2 drd = 4 integraldisplay 2 re r 2 dr . moseley (cmm3869) HW16 Gilbert (56380) 2 The presence of the term r now allows this last integral to be evaluated by the subsitution u = r 2 . For then I = 2 bracketleftBig e u bracketrightBig 4 = 2 (1 e 4 ) . 003 10.0 points The solid shown in lies inside the sphere x 2 + y 2 + z 2 = 9 and outside the cylinder x 2 + y 2 = 4 . Find the volume of the part of this solid lying above the xyplane. 1. volume = 5 5 2. volume = 5 5 3 3. volume = 5 5 3 4. volume = 5 5 5. volume = 10 5 3 6. volume = 10 5 3 correct Explanation: From directly overhead the solid is similar to x y 3 2 r In Cartesian coordinates this is the annulus R = braceleftBig ( x, y ) : 4 x 2 + y 2 9 bracerightBig . Thus the volume of the solid above the xy plane is given by the integral V = integraldisplay integraldisplay R (9 x 2 y 2 ) 1 / 2 dxdy . To evaluate V we change to polar coordi nates. Now R = braceleftBig ( r, ) : 2 r 3 , 2 bracerightBig , so that after changing coordinates the integral becomes V = integraldisplay 3 2 integraldisplay 2 radicalbig 9 r 2 rdrd = 2 integraldisplay 3 2 r radicalbig 9 r 2 dr = bracketleftBig 2 3 (9 u ) 3 / 2 bracketrightBig 9 4 , using the substitution u = r 2 . Consequently, volume =...
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This note was uploaded on 04/12/2011 for the course M 408d taught by Professor Sadler during the Spring '07 term at University of Texas at Austin.
 Spring '07
 Sadler
 Multivariable Calculus, Polar Coordinates

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