91Ó°ÊÓ

$S=\left\{\left(x,y\right)|x^2+y^2≤1,xâ‰\yen 0\right\}$

$S=\left\{\left(x,y\right)|x^2+y^2≤1,xâ‰\yen 0\right\}$

$\begin{array}{rcl}\stackrel{¯}{x}& =& \frac{{∫}_0^1{∫}_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}xkxdydx}{{∫}_0^1{∫}_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}kxdydx}\\ & =& \frac{2{∫}_0^1{∫}_0^{\sqrt{1-x^2}}k{x}^{2}dydx}{2{∫}_0^1{∫}_0^{\sqrt{1-x^2}}kxdydx}\end{array}$

Take $x=r\cos \mathrm{θ},y=r\sin \mathrm{θ}$

width="336">x¯=∫0π/2∫01kr2cos2θrdrdθ∫0π/2∫01krcosθrdrdθ=∫0π/2∫01r3cos2θdrdθ∫0π/2∫01r2cosθdrdθ=∫0π/2r4401cos2θdθ∫0π/2r3301cosθdθ=∫0π/2r4401cos2θdθ∫0π/2r3301cosθdθ=14∫0π/2cos2θdθ13∫0π/2cosθdθusecos2θ=1+cos2θ2=14∫0π/212(1+cos2θ)dθ13∫0π/2cosθdθ=18θ+12sin2θ0π/213[sinθ]0π/2=π16-sin2π213sinπ2-sin(0)=π16-013(1-0)=3π16

$\begin{array}{rcl}\stackrel{¯}{y}& =& \frac{{âˆ\neg }_{\mathrm{Î\copyright }}y\mathrm{ÒÏ}(x,y)dA}{{âˆ\neg }_{\mathrm{Î\copyright }}\mathrm{ÒÏ}(x,y)dA}\\ & =& \frac{{∫}_0^1{∫}_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}ykxdydx}{{∫}_0^1{∫}_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}kxdydx}\\ & =& \frac{{∫}_0^{\mathrm{Ï€}/2}{∫}_0^{1}r\sin \mathrm{θ}kr\cos \mathrm{θ}rdrd\mathrm{θ}}{{∫}_0^{\mathrm{Ï€}/2}{∫}_0^{1}kr\cos \mathrm{θ}rdrd\mathrm{θ}}\\ & =& \frac{{∫}_0^{\mathrm{Ï€}/2}{∫}_0^1{r}^3\sin \mathrm{θ}\cos \mathrm{θ}drd\mathrm{θ}}{{∫}_0^{\mathrm{Ï€}/2}{∫}_0^1{r}^2\cos \mathrm{θ}drd\mathrm{θ}}\\ & =& \frac{\frac{1}{2}{∫}_0^{\mathrm{Ï€}/2}{\left[\frac{r^4}{4}\right]}_0^1\sin 2\mathrm{θ}d\mathrm{θ}}{{∫}_0^{\mathrm{Ï€}/2}{\left[\frac{r^3}{3}\right]}_0^1\cos \mathrm{θ}d\mathrm{θ}}\\ & =& \frac{\frac{1}{8}{∫}_0^{\mathrm{Ï€}/2}\sin 2\mathrm{θ}d\mathrm{θ}}{\frac{1}{3}{∫}_0^{\mathrm{Ï€}/2}\cos \mathrm{θ}d\mathrm{θ}}\\ & =& \frac{\frac{1}{8}{\left[-\frac{1}{2}\cos 2\mathrm{θ}\right]}_0^{\mathrm{Ï€}/2}}{\frac{1}{3}[\sin \mathrm{θ}{]}_0^{\mathrm{Ï€}/2}}\\ & =& \frac{\frac{1}{8}\left[-\frac{1}{2}\cos 2\left(\frac{\mathrm{Ï€}}{2}\right)+\frac{1}{2}\cos \left(0\right)\right.}{\frac{1}{3}\left[\sin \left(\frac{\mathrm{Ï€}}{2}\right)-\sin \left(0\right)\right]}\\ & =& \frac{\left(\frac{1}{8}\right)}{\left(\frac{1}{3}\right)}\\ & =& \frac{3}{8}\end{array}$

Therefore, the center of mass of the region is $(\stackrel{¯}{x},\stackrel{¯}{y})=\left(\frac{3\mathrm{π}}{16},\frac{3}{8}\right)$