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Chapter 13: Q 41 (page 1066)

The iterated integrals in Exercises 39–42 use cylindrical coordinates. Describe the solids determined by the limits of integration.

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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.

Evaluate each of the double integrals in Exercises $37 - 54$ as iterated integrals.

role="math" localid="1650327788023" $\int \int_{R} y \sin x d A,$

whererole="math" localid="1650327080219" $R = \{(x, y) | 0 \leq x \leq \frac{π}{2} a n d 0 \leq y \leq 1\}$

Let $Ω$ be a lamina in the xy-plane. Suppose $Ω$ is composed of n non-overlapping laminæ role="math" localid="1650321722341" $Ω_{1}, Ω_{2}, \dots, Ω_{n} .$ Show that if the masses of these laminæ are $m_{1}, m_{2}, \dots, m_{n}$ and the centers of masses are $({\bar{x}}_{1}, {\bar{y}}_{1}), ({\bar{x}}_{2}, {\bar{y}}_{2}), \dots, ({\bar{x}}_{n}, {\bar{y}}_{n}),$ then the center of mass of $Ω$ is $(\bar{x}, \bar{y}),$ where

$\bar{x} = \frac{\sum_{k = 1}^{n} m_{k} {\bar{x}}_{k}}{\sum_{k = 1}^{n} m_{k}} and \bar{y} = \frac{\sum_{k = 1}^{n} m_{k} {\bar{y}}_{k}}{\sum_{k = 1}^{n} m_{k}} .$

Evaluate each of the double integral in the exercise 37-54 as iterated integrals

$\int \int_{R} (2 - 3 x^{2} + y^{3}) d A W h e r e R = {(x, y) | - 3 \leq x \leq 2, 3 \leq y \leq 5}$

In Exercises, let $S = \{(x, y) ∣ x^{2} + y^{2} \leq 1 and x \geq 0\} .$

If the density at each point in S is proportional to the point’s distance from the origin, find the center of mass of S.

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The iterated integrals in Exercises 39–42 use cylindrical coordinates. Describe the solids determined by the limits of integration.

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Step 1:Given information

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