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’s 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

Given Information

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

Calculation of Mass

The mass is calculated as $m = \iint_{Ω} ρ (x, y) d A$

and $ρ (x, y) = k x^{2}$

$m = \int_{0}^{b} \int_{0}^{h} k x^{2} d y d x$

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

$m = \int_{0}^{b} k x^{2} h d x = k h \int_{0}^{b} x^{2} d x$

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

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

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

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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’s distance from the $y$ -axis, find the mass of $R$ .

Short Answer

Step by step solution

Given Information

Calculation of Mass

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Let Rbe rectangular region of vertices (0,0),(b,0),(0,h),and(b,h)If the density at each point in Ris proportional to the square of the point’s distance from the y-axis, find the mass ofR.

Short Answer

Step by step solution

Given Information

Calculation of Mass

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Calculus

Logic and Functions

Statistics

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Discrete Mathematics

Study anywhere. Anytime. Across all devices.

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’s distance from the $y$ -axis, find the mass of $R$ .