Chapter 13: Q 53. (page 1039)

Let $S = \{(x, y) | x^{2} + y^{2} \leq 1, x \geq 0\}$
If the density at each point in S is proportional to the point’s distance from the y-axis, find the mass of S.

Short Answer

Expert verified

Answer is $\frac{2}{3} k, k$ is the constant of proportionality.

Step by step solution

Step 1. Given information

$S = \{(x, y) | x^{2} + y^{2} \leq 1, x \geq 0\}$

Step 2. Explanation

$S = \{(x, y) | x^{2} + y^{2} \leq 1, x \geq 0\}$

Mass $= \int_{0}^{1} \int_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} k x d y d x$

$= 2 \int_{0}^{1} \int_{0}^{\sqrt{1 - x^{2}}} k x d y d x$

Take $x = r \cos θ, y = r \sin θ$

role="math" localid="1653727977284" $\begin{array}{rcl} Mass & = & 2 \int_{0}^{\frac{π}{2}} \int_{0}^{1} k r \cos θ r d r d θ \\ = & 2 k \int_{0}^{\frac{π}{2}} \int_{0}^{1} r^{2} \cos θ d r d θ \\ = & 2 k \int_{0}^{\frac{π}{2}} {[\frac{r^{3}}{3}]}_{0}^{1} \cos θ d θ \\ = & \frac{2}{3} k {[\sin θ]}_{0}^{\frac{π}{2}} \\ = & \frac{2}{3} k [\sin (\frac{π}{2}) - \sin 0] \\ = & \frac{2}{3} k \end{array}$

Mass is equal to $\frac{2}{3} k, k$ is the constant of proportionality.

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 $S = \{(x, y) | x^{2} + y^{2} \leq 1, x \geq 0\}$
If the density at each point in S is proportional to the point’s distance from the y-axis, find the mass of S.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Applied Mathematics

Decision Maths

Probability and Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Let S=x,y|x2+y2≤1,x≥0If the density at each point in S is proportional to the point’s distance from the y-axis, find the mass of S.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Applied Mathematics

Decision Maths

Probability and Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Let $S = \{(x, y) | x^{2} + y^{2} \leq 1, x \geq 0\}$
If the density at each point in S is proportional to the point’s distance from the y-axis, find the mass of S.