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Unformatted text preview: MAC 2313, Fall 2010 Midterm 4 Review Problems Solutions  1 We will discuss these problems in class on Monday 11/15. Solutions will be posted on the course webpage over the weekend. 1 A bounded region in the first octant of 3dimensional Euclidean space has the surface x + y + z 2 = 1 as part of its boundary. The remainder of its boundary is given by portions of the places x = 0, y = 0, and z = 0. Compute the triple integral of z over this region in space. (You should probably begin by sketching this region.) Solution: The surface x + y + z 2 = 1 lies in the first octant for x 0, y 0, and x + y 1. So, we can express the integral as integraldisplay 1 integraldisplay 1 x integraldisplay 1 x y z dz dy dx = integraldisplay 1 integraldisplay 1 x bracketleftbigg z 2 2 bracketrightbigg z = 1 x y z =0 dy dx, = integraldisplay 1 integraldisplay 1 x 1 x y 2 dy dx, = integraldisplay 1 bracketleftbigg (1 x ) y 2 y 2 4 bracketrightbigg y =1 x y =0 dx, = integraldisplay 1 (1 x )(1 x ) 2 (1 x ) 2 4 dx, = integraldisplay 1 (1 x ) 2 4 dx, = integraldisplay 1 (1 2 x + x 2 4 dx, = bracketleftbigg x x 2 4 + x 3 12 bracketrightbigg x =1 x =0 = 1 12 . 2 Let R denote the portion of the unit ball { ( x, y, z ) : x 2 + y 2 + z 2 1 } that lies inside the solid cone { ( x, y, z ) : z radicalbig x 2 + y 2 } . Compute the volume of R with triple integrals in both cylindrical and spherical coordinates. Solution: In cylindrical coordinates, we first integrate z from the cone z = radicalbig x 2 + y 2 = r 2 = r to the top of the ball, z = radicalbig 1 x 2 y 2 = 1 r 2 . Then comes the trickiest bound: with respect to r . We need to determine at which value of r the top of the ball meets the bottom of the cone. This value of r satisfies 1 r 2 = r , so 1 r 2 = r 2 , so 2 r 2 = 1, so MAC 2313, Fall 2010 Midterm 4 Review Problems Solutions  2 r = radicalbig 1 / 2 . We integrate r from 0 to this value. Finally we integrate from 0 to 2 . This integral is therefore integraldisplay 2 integraldisplay 1 / 2 integraldisplay 1...
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This note was uploaded on 12/10/2011 for the course MAC 2313 taught by Professor Keeran during the Fall '08 term at University of Florida.
 Fall '08
 Keeran
 Calculus

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