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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.

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Most popular questions from this chapter

Use the results of Exercises 59 and 60 to find the centers of masses of the laminæ in Exercises 61–67.

Use the lamina from Exercise 61, but assume that the density is proportional to the distance from the x-axis.

Evaluate the triple integrals over the specified rectangular solid region.

$\begin{array}{r} ∭_{R} x^{2} y z d V, where \\ R = {(x, y, z) ∣ - 2 \leq x \leq 1, 0 \leq y \leq 3, and 1 \leq z \leq 5} \end{array}$

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.

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane

$f (x, y) = \sin x \cos y W h e r e R = {(x, y) | 0 \leq x \leq π, 0 \leq y \leq π}$

Evaluate the triple integrals over the specified rectangular solid region.

$\begin{aligned} ∭_{R} z \sin x \cos y d V, where \\ R = \{(x, y, z) ∣ 0 \leq x \leq π, \frac{3 π}{2} \leq y \leq 2 π, and 1 \leq z \leq 3\} \end{aligned}$

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Explain how to construct a Riemann sum for a function of two variables over a rectangular region.

Short Answer

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Step 1. Given information:

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