Chapter 13: Q 39. (page 1039)

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

Short Answer

Expert verified

The mass of $R$ is $m = \frac{1}{2} k h b^{2}$ .

Step by step solution

Given Information

The vertices of rectangular region are $(0, 0), (b, 0), (0, h) and (b, h)$

$ρ (x, y) = k x$

Calculation of Mass

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

$m = \int_{0}^{b} \int_{0}^{h} k x d y d x$ as $ρ (x, y) = k x$

Solving inner integral first $m = \int_{0}^{b} k x [y]_{0}^{h} d x$

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

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

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

$m = \frac{1}{2} k h b^{2}$

The required mass is $m = \frac{1}{2} k h b^{2}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

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

Statistics

Probability and Statistics

Theoretical and Mathematical Physics

Discrete Mathematics

Pure Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

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

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

Statistics

Probability and Statistics

Theoretical and Mathematical Physics

Discrete Mathematics

Pure Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

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