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Unformatted text preview: Solutions to the Practice Final Note: You may have noticed that none of the questions based on Chapter 16 had hints about how to approach them. Some of the questions on the actual final will have such hints, but in order to allow you to practice deciding how to approach such questions, I asked all of the questions on the practice final without hints. Hopefully this gave you a chance to try out some of the advice from the last lecture. 1. Consider the following integral Z 1 Z 2- 2 x Z 1- x- 1 2 y f ( x, y, z ) dz dy dx. (a) Rewrite this integral so that the inner most integral is with respect to z , the middle integral is with respect to x , and the outer integral is with respect to y . We see that the region of integration is bounded by the planes x = 0, y = 0, z = 0, and 2 x + y + 2 z = 2. Thus the integral can be re-written as Z 2 Z 1- 1 2 y Z 1- x- 1 2 y f ( x, y, z ) dz dx dy. (b) Rewrite this integral so that the inner most integral is with respect to x , the middle integral is with respect to y , and the outer integral is with respect to z . Similarly, we see that the integral can also be re-written as Z 1 Z 2- 2 z Z 1- z- 1 2 y f ( x, y, z ) dx dy dz. 2. Consider the part of the solid sphere of radius 2 contained in the first quadrant (that is, the set x 2 + y 2 + z 2 4 , x , y , and z ). Suppose it has density ( x, y, z ) = z . Compute its moment of inertia about the z-axis. Recall that, in spherical coordinates, x = sin cos , y = sin sin , z = cos and the Jacobian is 2 sin . In spherical coordinates, the region of integration is obtained by letting go from 0 to 2 and both and go from 0 to / 2. The moment of inertia is obtained by integrating x 2 + y 2 times the density over the solid. Writing this in spherical coordinates (and remembering the Jacobian), we see that the moment of inertia of this solid around the z-axis is given by Z 2 Z / 2 Z / 2 cos ( 2 sin 2 cos 2 + 2 sin 2 sin 2 ) 2 sin d d d = Z 2 Z / 2 Z / 2 5 cos sin 3 d d d = 1 6 6 2 1 4 sin 4 / 2 [ ] / 2 = 4 3 . 1 3. Suppose that Sarah takes two buses to work. At the first bus stop, she waits X minutes for the bus, where X has distribution function f ( x ) = 1 5 if x 5 otherwise . When she transfers to the second bus, she waits Y minutes for it, where Y has distribution function g ( y ) = 1 2 e- y/ 2 if y otherwise , and X and Y are independent. What is the probability that the total time Sarah spends waiting for buses is less than 4 minutes?...
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- Spring '07