Centroid Of An Area 
The centroid of an area is similar to the center of mass of a body. Calculating the centroid involves only the geometrical shape of the area. The center of gravity will equal the centroid if the body is homogenous i.e. constant density. Integration formulas for calculating the Centroid are: 
When calculating the centroid of a complex shape. Divide the shape up into a combination of known shapes. Then use the the following formula: 

The centroid location of many common shapes is known. The Properties of Areas page includes the Centroid Location, Area, Area Moments of Inertia, Area Polar Moments of Inertia, & Area Radius of Gyration for many common shapes. 