The centroid of a plane region $R$ is its geometric center $\left(\stackrel{\&conjugate0;}{x}\,\stackrel{\&conjugate0;}{y}\right)$. If a thin slab with uniform density and the shape of $R$ were placed horizontal, it would balance on a support placed under the centroid.

Let $R_x$ be a region $R$ determined by $f\left(x\right)\ge g\left(x\right)\,a\le x\le b$, and let $R_y$ be a region $R$ determined by  $u\left(y\right)\ge v\left(y\right)\,\mathrm{\α}\le y\le \mathrm{\β}$. Figures 5.7.1 and 5.7.2 are examples of regions $R_x$ and $R_y$, respectively.

If $R$ is a region $R_x$, then its centroid is defined as on the left in Table 5.7.1.

If $R$ is a region $R_y$, then its centroid is defined as on the right in Table 5.7.1.

where $A$ is the area of $R$.

The centroid of a plane curve $C$ is its geometric center $\left(\stackrel{\&conjugate0;}{x}\,\stackrel{\&conjugate0;}{y}\right)$. If $S$ is the arc length of the curve and $\mathrm{ds}$ the element of arc length, then the coordinates of the centroid are found via the formulas in Table 5.7.2.