Show that in polar coordinates the formulas for the centroid(x, y) of a region R arex =

Chapter 14, Problem 13

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Show that in polar coordinates the formulas for the centroid $(\overline{\mathrm{x}}, \overline{\mathrm{y}})$ of a region R are

$\bar{x}=\frac{1}{\text { area of } R} \iint_{R} r^{2} \cos \theta d r d \theta$

$\bar{y}=\frac{1}{\text { area of } R} \iint_{R} r^{2} \sin \theta d r d \theta$

Equation Transcription:

$\left({\overline{x},\overline{y}}\right)$

$\overline{x}=\frac{1}{area\ of\ R}\int\limits_{\ }^{\ }\int\limits_{R}^{\ }{r}^{2}cos\ \theta{}\ dr\ d\theta{}$

$\overline{y}=\frac{1}{area\ of\ R}\int\limits_{\ }^{\ }\int\limits_{R}^{\ }{r}^{2}sin\ \theta{}\ dr\ d\theta{}$

text transcription:

(x bar, y bar)

X bar = 1/area of R integral integral R r^2 cos theta dr d theta

Y bar = 1/area of R integral integral R r^2 sin theta dr d theta

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