How to find the centroid of a plane region

Formulas for the centroid

The centroid of a plane region is the center point of the region over the interval $[a,b]$. In order to calculate the coordinates of the centroid, we’ll need to calculate the area of the region first.

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The equation for this area is

$A=\int^b_af(x)\ dx$

To calculate the coordinates of the centroid $(\overline{x},\overline{y})$, we’ll use

$\overline{x}=\frac{1}{A}\int^b_axf(x)\ dx$

$\overline{y}=\frac{1}{A}\int^b_a\frac12\left[f(x)\right]^2\ dx$

What is the centroid, and how do we find its coordinates?

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Finding the centroid of a region bounded by specific curves

Example

Find the centroid of the region bounded by the curves $x=1$, $x=6$, $y=0$, and $y=4$.

First we’ll find the area of the region using

$A=\int^b_af(x)\ dx$

We can use the $x$-values as the boundaries of the interval, so $[a,b]$ is $[1,6]$. To find $f(x)$, we need to remember that taking the integral of a function is the same thing as finding the area underneath the function. Which means we treat this like an “area between curves” problem, and we get

$A=\int^6_14-0\ dx$

$A=4\int^6_1dx$

$A=4x\big|^6_1$

$A=4(6)-4(1)$

$A=20$

In order to calculate the coordinates of the centroid, we’ll need to calculate the area of the region first.

Now we can use the formulas for $\bar{x}$ and $\bar{y}$ to find the coordinates of the centroid. First, let’s solve for $\bar{x}$.

$\overline{x}=\frac{1}{A}\int^b_axf(x)\ dx$

$\overline{x}=\frac{1}{20}\int^b_ax(4-0)\ dx$

$\overline{x}=\frac{1}{5}\int^6_1x\ dx$

$\overline{x}=\frac15\left(\frac{x^2}{2}\right)\bigg|^6_1$

$\overline{x}=\frac{x^2}{10}\bigg|^6_1$

$\overline{x}=\frac{(6)^2}{10}-\frac{(1)^2}{10}$

$\overline{x}=\frac72$

Now we’ll find $\overline{y}$.

$\overline{y}=\frac{1}{A}\int^b_a\frac12\left[f(x)\right]^2\ dx$

$\overline{y}=\frac{1}{20}\int^b_a\frac12(4-0)^2\ dx$

$\overline{y}=\frac25\int^6_1dx$

$\overline{y}=\frac{2x}{5}\bigg|^6_1$

$\overline{y}=\frac{2(6)}{5}-\frac{2(1)}{5}$

$\overline{y}=2$

The centroid of the region is at the point $\left(\frac{7}{2},2\right)$.

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