Chapter 13: Q 47. (page 1039)

Let $C = \{(x, y) | x^{2} + y^{2} \leq 1\}$
If the density at each point in C is proportional to the point’s 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

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

Step 2. Explanation

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

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

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

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Let $C = \{(x, y) | x^{2} + y^{2} \leq 1\}$
If the density at each point in C is proportional to the point’s 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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Let C=(x,y)|x2+y2≤1If the density at each point in C is proportional to the point’s 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

Probability and Statistics

Statistics

Decision Maths

Theoretical and Mathematical Physics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

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