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Chapter 13: Q 42. (page 1039)

Let $R$ be rectangular region of vertices $(0, 0), (b, 0), (0, h), and (b, h)$
If the density at each point in $R$ is proportional to the square of the point鈥檚 distance from the $y$ -axis, find the mass of $R$ .

Short Answer

Expert verified

The mass is $m = \frac{1}{3} k h b^{3}$ .

Step by step solution

01

Given Information

Vertices of rectangular region is $(0, 0), (b, 0), (0, h) and (b, h)$

02

Calculation of Mass

The mass is calculated as $m = {鈭�}_{惟} 蚁 (x, y) d A$

and $蚁 (x, y) = k x^{2}$

$m = {鈭�}_{0}^{b} {鈭�}_{0}^{h} k x^{2} d y d x$

$m = {鈭�}_{0}^{b} k x^{2} [y]_{0}^{h} d x$

$m = {鈭�}_{0}^{b} k x^{2} h d x = k h {鈭�}_{0}^{b} x^{2} d x$

$鈬� m = k h {[\frac{x^{3}}{3}]}_{0}^{b}$

$m = \frac{1}{3} k h b^{3}$

The mass is $m = \frac{1}{3} k h b^{3}$

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Most popular questions from this chapter

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.

${鈭�}_{0}^{3} {鈭�}_{0}^{3} {鈭�}_{0}^{3 - y} f (x, y, z) d z d y d x$

$Express the sum \frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} using double - summation notation .$

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

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

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

$\overset{炉}{x} = \frac{{鈭�}_{k = 1}^{n} m_{k} {\overset{炉}{x}}_{k}}{{鈭�}_{k = 1}^{n} m_{k}} and \overset{炉}{y} = \frac{{鈭�}_{k = 1}^{n} m_{k} {\overset{炉}{y}}_{k}}{{鈭�}_{k = 1}^{n} m_{k}} .$

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

role="math" localid="1650327788023" $鈭� {鈭�}_{R} y \sin x d A,$

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

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