Chapter 13: Q 56. (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 origin, find the mass of S.

Short Answer

Expert verified

Answer is $\begin{array}{rcl} \frac{π}{3} k \end{array}$ where 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

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

$\begin{array}{rcl} m & = & \int_{0}^{1} \int_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} k \sqrt{x^{2} + y^{2}} d y d x \\ = & 2 \int_{0}^{1} \int_{0}^{\sqrt{1 - x^{2}}} k \sqrt{x^{2} + y^{2}} d y d x \end{array}$

Let $x = r \cos θ, y = r \sin θ$

$\begin{array}{rcl} m & = & 2 k \int_{0}^{π / 2} {[\frac{r^{3}}{3}]}_{0}^{1} d θ \\ = & 2 k \int_{0}^{π / 2} [\frac{1}{3} - \frac{0}{3}] d θ \\ = & \frac{2}{3} k \int_{0}^{π / 2} d θ \\ = & \frac{2}{3} k [θ]_{0}^{π / 2} \\ = & \frac{2}{3} k [\frac{π}{2} - 0] \\ = & \frac{π}{3} k \end{array}$

k is the constant of proportionality.

Mass is 9 units.

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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 origin, find the mass of S.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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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 origin, 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

Logic and Functions

Pure Maths

Statistics

Decision Maths

Discrete Mathematics

Applied Mathematics

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 origin, find the mass of S.