Chapter 13: Q. 29 (page 1083)
Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.
Chapter 13: Q. 29 (page 1083)
Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.
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Get started for freeUse the lamina from Exercise 64, but assume that the density is proportional to the distance from the x-axis.
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 at each point in Ris proportional to the distance of the point from the xy-plane.
(a) Without using calculus, explain why the x- and y-coordinates of the center of mass are respectively.
(b) Use an appropriate integral expression to find the z-coordinate of the center of mass.
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 how to construct a midpoint Riemann sum for a function of three variables over a rectangular solid for which each is the midpoint of the subsolid role="math" localid="1650346869585" . Refer either to your answer to Exercise or to Definition .
Describe the three-dimensional region expressed in each iterated integral in Exercises 35–44.
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