22M:28
Spring 05
J. Simon
Study Guide for Final Exam
Chapter 5 Portion
[See comments/suggestions/"how to use..." on earlier handouts]
In addition to the problems below, here are appropriate study problems from the Miscellaneous Exercises for Chapter 5, page 371
Problems
1, 2, 5, 6,
10 (Hints: (a) The new limits of integration should be a rectangle in (u,v) space:
u=0..1, v=0..6
(b) The numerical answer is 108 e - 108 .
17
Problem 5.1 Evaluate the iterated integrals and draw the domain of integration.
(a)
Answer: 484/5 .
(b)
Answer: 14/3
Problem 5.2 Here is a contour plot of a function f(x,y) on the domain
D given by x = 0..2 , y = 0..2. Use this data to estimate .
(Note the contours correspond to values of f(x,y) = 0, 2, 4, 6, 8, 10, 12, 14; but there wasn't room to label the last few in the picture.)
Problem 5.3
Integrate the function f(x,y) = x - xy over the region bounded by the curves/lines
x + y = 1 and y = 1 - x2 .
Answer: 1/24 .
Remark: I might give you more complicated integrands and just ask you to set up an iterated integral whose value is the desired double integral. Of course, we can consider many particular regions in the plane.
Problem 5.4 Find the volume of the solid region in the first octant bounded by the surface z = 16 - x2 - y2 .
Answer: 32 p » 100.5 (Hint: It is *much* easier to evaluate this integral in polar coordinates than in cartesian coordinates.)
Problem 5.5 Express as an equivalent iterated integral with the order of integration reversed.
Remark: You should make up a number of similar examples to practice this process.
Problem 5.6
(a) Show that the "stretching factor" for the change of variables
x = r cos(t) , y = r sin(t) is r .
(Hint: Write out the Jacobian matrix and evaluate its determinant.)
(b) Calculate the "stretching factor" for the change of variables
x = 3u , y=ev .
(c) We want to evaluate the integral of f(x,y) = xy on the parallelogram domain bounded by the lines
Use the change of variables u = x+2y , v = y-x to set up (but do not evaluate) an iterated integral in terms of u and v that has the correct value.
Evaluate the iterated integral in u,v that you just wrote. Based on the overall behavior of the function f(x,y) on the given domain, should your integral come out positive or negative? Did you get it right?? (Answer: 106/81)
Problem 5.7. (This could be several problems, or one, depending on whether you are asked to evaluate the integral(s) or just set it/them up.)
[Based on Misc. Exercise 16, page 374]
Suppose an object is a hollow ball, i.e. "spherical shell", i.e the region between two concentric spheres. The outer sphere has radius 2 and the inner sphere has radius 1. Suppose the density of the object varies, with density at each point = 5 r2 , where r is the distance from a point to the center of the spheres. Locate the object so its center is the origin in R3.
(a) Set up, but do not evaluate, an iterated integral whose value is the mass of the object.
(b) Evaluate your integral from part (a). [Answer: 124 p .
(b) Cut the object in half horizontally, keep the upper half and discard the lower half. Find the center of mass of the new object. [Take advantage of any symmetries you see to minimize your work.] Answer: (0,0, 105/124).
Problem 5.8
Problems similar to Problem 5.7:
(a) The given object is the region between two concentric circular cylinders of radii 1 and 2, with height = 5, and density function equal to 3 times the distance from a point to the central axis. Find the total mass (Answer 70 p). Then cut the object in half vertically to make a new object (that looks like a thick trough standing on end; all points in the original object with y ³ 0; find it's center of mass. [Answer ].
(b) The given object is the region between the planes z = x+y , and z = 2x + 2y , for
x = 0..5, y=0..5. Density d(x,y,z) = z . Find the total mass. (Ans. 4375/4). Find the coordinates of the center of mass (ok to leave your answers in terms of one or more iterated integrals; evaluating the integral for total mass is enough evaluation of integrals for this problem).[Answer ( 45/14, 45/14, 10 ) ].
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