Answered: Find the centroid of the region in the…

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter7: Locus And Concurrence

Section7.2: Concurrence Of Lines

Problem 7E: Which lines or line segments or rays must be drawn or constructed in a triangle to locate its a...

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Related questions

Question

Find the centroid of the region in the first quadrant
bounded by the rays θ = 0 and θ = π/2 and the circles r = 1
and r = 3.

Expert Solution

Step 1

Given:

In the first quadrant area bounded by the rays are θ = 0 and θ = $\frac{π}{2}$ ,

And the circles r = 1 and r = 3.

To find:

The centroid of the region in the first quadrant.

Step 2

So, Find M such that,

$M = \int_{0}^{\frac{π}{2}} \int_{1}^{3} r d r d θ$ (1)

As θ = 0, θ = $\frac{π}{2}$ , and r = 1, r = 3.

Integrate equation with respect to r,

$M = \int_{0}^{\frac{π}{2}} {[\frac{r^{2}}{2}]}_{1}^{3} d θ$

$M = \int_{0}^{\frac{π}{2}} [\frac{3^{2}}{2} - \frac{1}{2}] d θ$

$M = 4 \int_{0}^{\frac{π}{2}} d θ$

Again integrate with respect to $θ$ ,

$M = 4 {[x]}_{0}^{\frac{π}{2}}$

Thus, $M = 2 π$

Step by step

Solved in 3 steps

Similar questions

Which lines or line segments or rays must be drawn or constructed in a triangle to locate its a orthocenter? b centroid?

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Find the centroid of the region in the first quadrant bounded by the rays θ = 0 and θ = π/2 and the circles r = 1 and r = 3.