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Chapter 13: Q 38 (page 1066)
Describe the graphs of the equations and provide alternative equations in the specified coordinate systems.
. Change to the rectangular and cylindrical systems.
The equation in rectangular coordinates
The equation in cylindrical coordinates
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Let be a lamina in the xy-plane. Suppose is composed of two non-overlapping lamin and , as follows:
Show that if the masses and centers of masses of and are and and respectively, then the center of mass of is where
Explain how to construct a midpoint Riemann sum for a function of two variables over a rectangular region for which each is the midpoint of the subrectangle
Refer to your answer to Exercise 10 or to Definition 13.3.
In the following lamina, all angles are right angles and the density is constant:
Let be a continuous function of three variables, let localid="1650352548375" be a set of points in the -plane, and let localid="1650354983053" be a set of points in -space. Find an iterated triple integral equal to the triple integral localid="1650353288865" . How would your answer change iflocalid="1650352747263" ?
In Exercises 57鈥�60, let R be the rectangular solid defined by
R = {(x, y, z) | 0 鈮� x 鈮� 4, 0 鈮� y 鈮� 3, 0 鈮� z 鈮� 2}.
Assume that the density ofR is uniform throughout.
(a) Without using calculus, explain why the center of mass is (2, 3/2, 1).
(b) Verify that the center of mass is (2, 3/2, 1), using the appropriate integral expressions.
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