Centroid

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The term centroid is used to describe the center of mass $\left(\bar{x},\bar{y}\right)$ of a flat plate (a lamina) with uniform density that occupies a region $R$ in the plane. In this case, the density function $\rho <!--FUNCTION\; APPLICATION-->\left(x,y\right)$ is constant, and consequently, it drops out of the computation for $\bar{x}$ and $\bar{y}$.

1. Find the centroid of the region bounded by the graph of $y=1/x$ and the line $2x+2y=5$.
   ${\displaystyle \left(\bar{x},\bar{y}\right)=\left(\frac{9}{30-32\ln <!--FUNCTION\; APPLICATION-->\left(2\right)},\frac{9}{30-32\ln <!--FUNCTION\; APPLICATION-->\left(2\right)}\right)}$
2. Find the centroid of the region obtained by connecting the points $\left(0,0\right)$, $\left(4,0\right)$, $\left(4,4\right)$, $\left(2,4\right)$, $\left(2,1\right)$, $\left(0,1\right)$, and $\left(0,0\right)$ in order by line segments.
   ${\displaystyle \left(\bar{x},\bar{y}\right)=\left(\frac{13}{5},\frac{17}{10}\right)}$
3. Find the centroid of the region bounded by the graphs of $y=x^2$ and $y=x^3$.
   ${\displaystyle \left(\bar{x},\bar{y}\right)=\left(\frac{3}{5},\frac{12}{35}\right)}$
4. Let $R$ be the region in the first quadrant that is inside the circle of radius $a$ centered at the origin. Find its centroid.
   ${\displaystyle \left(\bar{x},\bar{y}\right)=\left(\frac{4a}{3\pi },\frac{4a}{3\pi }\right)}$