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 freeLet be a continuous function of three variables, let be a set of points in the -plane, and let be a set of points in 3-space. Find an iterated triple integral equal to the the triple integral. How would your answer change if?
Find the masses of the solids described in Exercises 53–56.
The solid bounded above by the paraboloid with equation and bounded below by the rectangle in the xy-plane if the density at each point is proportional to the square of the distance of the point from the origin.
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 .
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.
Discuss the similarities and differences between the definition of the double integral found in Section and the definition of the triple integral found in this section.
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