Gilbert_Hmwk16sol - moseley (cmm3869) HW16 Gilbert (56380)...

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Unformatted text preview: moseley (cmm3869) HW16 Gilbert (56380) 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 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 xy-plane. 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 xy-plane bounded by the graph of x = radicalbig 4- y 2 and the y-axis. 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 xy-plane. 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.

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Gilbert_Hmwk16sol - moseley (cmm3869) HW16 Gilbert (56380)...

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