Q 49. Let C=x,y|x2+y2鈮�1If the densit... [FREE SOLUTION]

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Chapter 13: Q 49. (page 1039)

Let $C = \{(x, y) | x^{2} + y^{2} 鈮� 1\}$
If the density at each point in C is proportional to the point鈥檚 distance from the origin, find the mass of C.

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information

$C = \{(x, y) | x^{2} + y^{2} 鈮� 1\}$

Step 2. Explanation

Mass of lamina is equal to $鈭� 蚁 (x, y) d A$

Here, $蚁 (x, y)$ is the uniform density of the lamina.

As $蚁 (x, y)$ is proportional to point's distance from $y -$ axis, take $蚁 (x, y) = k \sqrt{x^{2} + y^{2}}$

So,

$\begin{array}{rcl} m & = & {鈭�}_{- 1}^{1} {鈭�}_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} k \sqrt{x^{2} + y^{2}} d y d x \\ = & 2 k {鈭�}_{- 1}^{1} {[\frac{1}{2} y \sqrt{x^{2} + y^{2}} + \frac{1}{2} x^{2} \ln (y + \sqrt{x^{2} + y^{2}})]}_{0}^{\sqrt{1 - x^{2}}} d x \\ = & 2 k {鈭�}_{- 1}^{1} [\frac{1}{2} \sqrt{1 - x^{2}} + \frac{1}{2} x^{2} \ln (\frac{1 + \sqrt{1 - x^{2}}}{x})] d x \\ = & 2 k {[\frac{x}{2} \sqrt{1 - x^{2}} + x \ln (\frac{1 + \sqrt{1 - x^{2}}}{x}) + \frac{3}{2} \sin^{- 1} x]}_{- 1}^{1} \\ = & \frac{2}{3} k 蟺 \end{array}$

k is the constant of proportionality.

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91影视

Let $C = \{(x, y) | x^{2} + y^{2} 鈮� 1\}$
If the density at each point in C is proportional to the point鈥檚 distance from the origin, find the mass of C.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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91影视

Let C=x,y|x2+y2鈮�1If the density at each point in C is proportional to the point鈥檚 distance from the origin, find the mass of C.

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

Mechanics Maths

Calculus

Discrete Mathematics

Decision Maths

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

Let $C = \{(x, y) | x^{2} + y^{2} 鈮� 1\}$
If the density at each point in C is proportional to the point鈥檚 distance from the origin, find the mass of C.