91影视

The given statement is the centroid of a circle is the center of the circle.

Consider a circle $x^2+y^2=a^2$of constant density with a center at $(0,0).$

xcoordinate of the centroid of the circle is following.

$$\begin{gathered} \stackrel{炉}{x}=\frac{鈭�x蚁(x,y)dxdy}{鈭�蚁(x,y)dxdy} \\ \stackrel{炉}{x}=\frac{{鈭�}_{x=鈭�a}^a{鈭�}_{y=鈭�\sqrt{a^2鈭�x^2}}^{\sqrt{a^2鈭�x^2}}xdxdy}{{鈭�}_{x=鈭�a}^a{鈭�}_{y=鈭�\sqrt{a^2鈭�x^2}}^{\sqrt{a^2鈭�x^2}}{\displaystyle d}{\displaystyle x}{\displaystyle d}{\displaystyle y}} \end{gathered}$$

change the system into a polar system by substituting $x=r\cos 胃,y=r\sin 胃\&dxdy=rdrd胃.$

localid="1650439208659" $$\begin{gathered} \stackrel{炉}{x}=\frac{{鈭�}_{胃=0}^{2蟺}{鈭�}_{胃=0}^a(r\cos 胃)rdrd胃}{{鈭�}_{胃=0}^{2蟺}{鈭�}_{r=0}^{a}rdrd胃} \\ \stackrel{炉}{x}=\frac{{\displaystyle \left({鈭�}_{r=0}^a{r}^{2}dr\right)}脳\left({鈭�}_{胃=0}^{2蟺}\cos 胃d胃\right)}{\left({鈭�}_{r=0}^a{r}^{2}dr\right)脳\left({鈭�}_{胃=0}^{2蟺}d胃\right)} \\ \stackrel{炉}{x}=\frac{\left(\frac{a^3}{3}\right)脳\left(0\right)}{\left(\frac{a^3}{3}\right)脳\left(2蟺\right)} \\ \stackrel{炉}{x}=0 \end{gathered}$$

ycoordinate of the centroid of the circle is following.

$$\begin{gathered} \stackrel{炉}{y}=\frac{鈭�y蚁(x,y)dxdy}{鈭�蚁(x,y)dxdy} \\ \stackrel{炉}{y}=\frac{{鈭�}_{x=鈭�a}^a{鈭�}_{y=鈭�\sqrt{a^2鈭�x^2}}^{\sqrt{a^2鈭�x^2}}ydxdy}{{鈭�}_{x=鈭�a}^a{鈭�}_{y=鈭�\sqrt{a^2鈭�x^2}}^{\sqrt{a^2鈭�x^2}}{\displaystyle }{\displaystyle d}{\displaystyle x}{\displaystyle d}{\displaystyle y}} \end{gathered}$$

change the system into a polar system by substituting$x=r\cos 胃,y=r\sin 胃\&dxdy=rdrd胃$

$\begin{array}{rl}\stackrel{炉}{y}& =\frac{{鈭�}_{胃=0}^{2蟺}{鈭�}_{r=0}^a\left(r\sin 胃\right)rdrd胃}{{鈭�}_{胃=0}^{2蟺}{鈭�}_{r=0}^{a}rdrd胃}\\ \stackrel{炉}{y}& =\frac{{鈭�}_{r=0}^a{r}^{2}dr脳{鈭�}_{胃=0}^{2蟺}\sin 胃d胃}{{鈭�}_{r=0}^a{r}^{2}dr脳{鈭�}_{胃=0}^{2蟺}d胃}\\ \stackrel{炉}{y}& =\frac{\left(\frac{a^3}{3}\right)脳\left(0\right)}{\left(\frac{a^3}{3}\right)脳\left(2蟺\right)}\\ \stackrel{炉}{y}& =0\end{array}$

the centroid of the circle is $p(\overline{x},\overline{y})=(0,0).$

So the centroid of a circle is the center of the circle.