**Centroid of Area:** The centroid of an area of arbitrary shape, such as the one shown below, can be determined using the integral equations

(1) |

where
*dA*= differential area and
*x y z*= moment arms of *dA* (identifying the coordinates of the centroid of dA, see figure)

Depending on the choice of differential area, we can have a single or a double integral in the numerator and denominator of each centroidal coordinate in Eq. (1) as demonstrated in the following examples.

Example 2 (updated)

If the surface being considered has boundaries that allow it to be represented by a collection of simple shapes, then a modified version of Eq. (1) can be used. For such composite shapes, as shown in the figure below, we find the location of centroid as

(2) |

where x_{i}, y_{i}, z_{i} denote the centroidal coordinates of the i^{th} element of the composite shape.

A composite shape should be subdivided in such a way as to simplify the calculation of centroidal location. In some cases it may be easier to think of the composite shape as a larger surface with a portion of it cut out. In the summation equation the area of the cutout portion would have a negative sign associated with it. The calculation of centroidal coordinates is demonstrated in the following example.