Q 52. Let S=x,y|x2+y2鈮�1,x鈮�0Find th... [FREE SOLUTION]

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Chapter 13: Q 52. (page 1039)

Let $S = \{(x, y) | x^{2} + y^{2} 鈮� 1, x 鈮� 0\}$
Find the centroid of S.

Short Answer

Expert verified

Answer is $(\frac{4}{3} 蟺, 0)$

Step by step solution

Step 1. Given information

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

Step 2. Explanation

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

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

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

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91影视

Let $S = \{(x, y) | x^{2} + y^{2} 鈮� 1, x 鈮� 0\}$
Find the centroid of S.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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91影视

Let S=x,y|x2+y2鈮�1,x鈮�0Find the centroid 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

Discrete Mathematics

Pure Maths

Mechanics Maths

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

Let $S = \{(x, y) | x^{2} + y^{2} 鈮� 1, x 鈮� 0\}$
Find the centroid of S.