At the end of this section you should be able to:

1. Find area and centroid of 2D shape defined by a function using calculus

2. Find area/volume and centroid of composite 2D/3D shape using the tabular method

   • Calculate centroid and area of common shapes

   • Find the center of mass of non-homogeneous objects

3. Simplify distributed loading into point load(s) in correct location(s)

This section teaches how to locate centroids and centers of mass and how to replace distributed loads with equivalent point loads for statics and internal load modeling. You will find the area and centroid of a 2D region defined by a function using calculus (integration with strips), and you will find the centroid of composite shapes in 2D and 3D using the tabular method, including shapes with holes or cutouts. You will extend the same tabular approach to center of mass for nonhomogeneous objects, where different parts have different densities or weights. You will also simplify common distributed loading cases into an equivalent resultant force located at the correct line of action, including uniform, triangular, and functionally defined distributed loads used in beam equilibrium. Mathematica resources are included to support setup, computation, and checking.