Chapter 13: Q 45. (page 1039)

$C = \{(x, y) ∣ x^{2} + y^{2} \leq 1\}$
Find the centroid of $C$ .

Short Answer

Expert verified

The centroid is $\bar{x} = 0, \bar{y} = 0$ .

Step by step solution

Given Information

It is given that $C = \{(x, y) ∣ x^{2} + y^{2} \leq 1\}$

Calculation of x-

The formula is

\bar{x} = \frac{\iint_{Ω} x ρ (x, y) d A}{\iint_{Ω} ρ (x, y) d A} and \bar{y} = \frac{\iint_{Ω} y ρ (x, y) d A}{\iint_{Ω} ρ (x, y) d A}

As density is uniform $ρ (x, y) = 1$

$\bar{x} = \frac{\int_{- 1}^{1} \int_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} x d y d x}{\int_{- 1}^{1} \int_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} d y d x} = \frac{\int_{- 1}^{1} x [y]_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} d x}{\int_{- 1}^{1} [y]_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} d x}$

$\bar{x} = \frac{\int_{- 1}^{1} x [\sqrt{1 - x^{2}} - (- \sqrt{1 - x^{2}})]}{\int_{- 1}^{1} [\sqrt{1 - x^{2}} - (- \sqrt{1 - x^{2}})] d x} = \frac{2 \int_{- 1}^{1} x \sqrt{1 - x^{2}}}{2 \int_{- 1}^{1} \sqrt{1 - x^{2}} d x}$

Region is symmetric about $y$ axis

$\bar{x} = \frac{0}{\frac{π}{2}} [\int_{- a}^{a} f (x) d x = 0, if f (x) is odd]$

$\bar{x} = 0$

Calculation of y-

Similarly $\bar{y} = \frac{\iint_{Ω} y ρ (x, y) d A}{\iint_{Ω} ρ (x, y) d A}$

$\bar{y} = \frac{\int_{- 1}^{1} \int_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} y d y d x}{\int_{- 1}^{1} \int_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} d y d x}$

$\bar{y} = \frac{\int_{- 1}^{1} [0] d x}{\int_{- 1}^{1} [y]_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} d x}$

$\bar{y} = 0$

The centroid is $\bar{x} = 0, \bar{y} = 0$

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$C = \{(x, y) ∣ x^{2} + y^{2} \leq 1\}$
Find the centroid of $C$ .

Short Answer

Step by step solution

Given Information

Calculation of x-

Calculation of y-

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C=(x,y)∣x2+y2≤1Find the centroid of C.

Short Answer

Step by step solution

Given Information

Calculation of x-

Calculation of y-

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Theoretical and Mathematical Physics

Statistics

Calculus

Probability and Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

$C = \{(x, y) ∣ x^{2} + y^{2} \leq 1\}$
Find the centroid of $C$ .