# 3.3.2: Aproximate Center of Area

*Fig. 3.3 Thin body center of mass/are schematic.*

In the previous case, the body was a three dimensional shape. There are cases where the body can be approximated as a two-dimensional shape because the body is with a thin with uniform density. Consider a uniform thin body with constant thickness shown in Figure 3.3 which has density, \(\rho\). Thus, equation 9 can be transferred into \[\bar{x} = \frac{1}{tA\rho}\int_{V} x \rho t dA \tag{11}\] The density, \(\rho\) and the thickness, \(t\), are constant and can be canceled. Thus equation 11 can be transferred into

Approximate \(x_i\) of Center Mass

\[\bar{x_{i}} = \frac{1}{A}\int_{A} x_{i} dA \tag{12}\]

when the integral now over only the area as oppose over the volume. Finding the centroid location should be done in the most convenient coordinate system since the location is coordinate independent.

### Contributors

Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.