Q 47. Let C=(x,y)|聽x2+y2鈮�1If the de... [FREE SOLUTION]

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Chapter 13: Q 47. (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 y-axis, find the center of mass of C.

Short Answer

Expert verified

Answer is $(0, 0)$

Step by step solution

Step 1. Given information

$C = \{(x, y) | x^{2} + y^{2} 鈮� 1\}$

Step 2. Explanation

$C = \{(x, y) | x^{2} + y^{2} 鈮� 1\}$

$\begin{array}{rcl} x & = & \frac{{鈭�}_{- 1}^{1} {鈭�}_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} x k x d y d x}{{鈭�}_{- 1}^{1} {鈭�}_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} k x d y d x} \\ = & \frac{4 {鈭�}_{0}^{1} {鈭�}_{0}^{\sqrt{1 - x^{2}}} x k x d y d x}{4 {鈭�}_{0}^{1} {鈭�}_{1}^{\sqrt{1 - x^{2}}} k x d y d x} \\ = & \frac{{鈭�}_{0}^{1} k x^{2} {[y]}_{0}^{\sqrt{1 - x^{2}}} d y d x}{{鈭�}_{0}^{1} k x {[y]}_{0}^{\sqrt{1 - x^{2}}} d y d x} \\ = & \frac{{鈭�}_{0}^{1} k x^{2} \sqrt{1 - x^{2}} d y d x}{{鈭�}_{0}^{1} k x \sqrt{1 - x^{2}} d y d x} \\ = & \frac{\frac{1}{8} {[x \sqrt{1 - x^{2}} (2 x^{2} - 1) + \sin^{- 1} x]}_{0}^{1}}{\frac{4}{3}} \\ = & \frac{0}{\frac{4}{3}} \\ = & 0 \end{array}$

$\begin{array}{rcl} y & = & \frac{{鈭�}_{- 1}^{1} {鈭�}_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} y k x d y d x}{{鈭�}_{- 1}^{1} {鈭�}_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} k x d y d x} \\ = & \frac{{鈭�}_{- 1}^{1} kx {[\frac{y^{2}}{2}]}_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}}}{{鈭�}_{- 1}^{1} kx {[y]}_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}}} \\ = & 0 \end{array}$

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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 y-axis, find the center of 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 y-axis, find the center of 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

Pure Maths

Calculus

Logic and Functions

Decision Maths

Applied Mathematics

Geometry

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 y-axis, find the center of mass of C.