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Chapter 13: Q. 10 (page 1003)
Explain how to construct a Riemann sum for a function of two variables over a rectangular region.
Ans:
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Let be an integrable function on the rectangular solid , and let Use the definition of the triple integral to prove that:
Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.
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.
Evaluate the sums in Exercises 23鈥�28.
In Exercises, let
If the density at each point in S is proportional to the point鈥檚 distance from the origin, find the moments of inertia about the x-axis, the y-axis, and the origin. Use these answers to find the radii of gyration of S about the x-axis, the y-axis, and the origin.
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