This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: romasko (qrr58) Assignment 11 luecke (55035) 1 This printout should have 9 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 10.0 points Hint: The volume under the graph of f(x,y) and over a region D is the integral of f(x,y) over D. The solid shown in is bounded by the paraboloid z = 4 1 4 ( x 2 + y 2 ) and the xyplane. Find the volume of this solid. 1. volume = 16 2. volume = 8 3. volume = 32 correct 4. volume = 8 5. volume = 16 6. volume = 32 Explanation: The paraboloid intersects the xyplane when z = 0, i.e. , when x 2 + y 2 16 = 0 . Thus the solid lies below the graph of z = 4 1 4 ( x 2 + y 2 ) and above the disk D = braceleftBig ( x, y ) : x 2 + y 2 16 bracerightBig , so its volume is given by the integral V = integraldisplay integraldisplay D parenleftBig 4 1 4 ( x 2 + y 2 ) parenrightBig dxdy . In polar coordinates this becomes V = 1 4 integraldisplay 4 integraldisplay 2 r (16 r 2 ) ddr = 1 2 integraldisplay 4 (16 r r 3 ) dr = 1 2 bracketleftBig 8 r 2 r 4 4 bracketrightBig 4 . Consequently, volume = V = 32 . 002 10.0 points Evaluate the integral I = integraldisplay integraldisplay R 4 e x 2 y 2 dxdy when R is the region in the first quadrant of the xyplane inside the graph of x = radicalbig 16 y 2 . 1. I = 2 (1 e 4 ) 2. I = 2 (1 e 16 ) 3. I = 4 (1 e 4 ) 4. I = 4 (1 e 16 ) romasko (qrr58) Assignment 11 luecke (55035) 2 5. I = (1 e 4 ) 6. I = (1 e 16 ) correct Explanation: In polar cooordinates, R is the set braceleftBig ( r, ) : 0 r 4 , 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 4 integraldisplay / 2 re r 2 ddr = 2 integraldisplay 4 re r 2 dr . The presence of the term r now allows this last integral to be evaluated by the subsitution u = r 2 . For then I = bracketleftBig e u bracketrightBig 16 = (1 e 16 ) . 003 10.0 points Which, if any, of the following are correct?...
View
Full
Document
 Fall '07
 Gilbert

Click to edit the document details