Chapter 13: Q.14 (page 991)
Show that when the density of the region is constant, the first moment about the -axis is
Short Answer
The first moment of the mass in about the axis is
Chapter 13: Q.14 (page 991)
Show that when the density of the region is constant, the first moment about the -axis is
The first moment of the mass in about the axis is
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Get started for freeIn Exercises 57–60, let R be the rectangular solid defined by
R = {(x, y, z) | 0 ≤ x ≤ 4, 0 ≤ y ≤ 3, 0 ≤ z ≤ 2}.
Assume that the density of R is uniform throughout, and find the moment of inertia about the x-axis and the radius of gyration about the x-axis.
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
In Exercises 57–60, let R be the rectangular solid defined by
R = {(x, y, z) | 0 ≤ x ≤ 4, 0 ≤ y ≤ 3, 0 ≤ z ≤ 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.
Explain how to construct a Riemann sum for a function of three variables over a rectangular solid.
Describe the three-dimensional region expressed in each iterated integral in Exercises 35–44.
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