Chapter 13: Q.13 (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.13 (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 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.
Evaluate each of the double integrals in Exercisesas iterated integrals.
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Identify the quantities determined by the integral expressions in Exercises 19–24. If x, y, and z are all measured in centimeters and ρ(x, y,z) is a density function in grams per cubic centimeter on the three-dimensional region , give the units of the expression.
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}.
Assuming that the density at each point in R is proportional to the distance of the point from the xy-plane, find the moment of inertia about the x-axis and the radius of gyration about the x-axis.
Explain why it would be difficult to evaluate the double integrals in Exercises 18 and 19 as iterated integrals.
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