center of mass

The center of mass of a body is the single point at which all of its mass could be imagined concentrated without changing how it balances or how it responds to a force. Balance a flat plate on a pin exactly under this point and it sits level; throw a wrench spinning across a room and this point traces a smooth parabola while the rest of the wrench tumbles around it. It is the mass-weighted average position of all the little pieces of the object.

Multiple integrals are precisely how you find it when the body has a continuous, possibly variable, density. For a planar lamina with surface density rho(x, y) over a region R, first compute the total mass M as the double integral over R of rho dA. Then the coordinates of the center of mass are mass-weighted averages: x-bar equals (1/M) times the double integral over R of x rho dA, and y-bar equals (1/M) times the double integral over R of y rho dA. The integrals in the numerators are the moments about the axes. For a solid body the same formulas use triple integrals and a volume density rho(x, y, z), giving three coordinates. When the density is constant it cancels out of the ratio, and the center of mass reduces to the centroid — the purely geometric average position of the region.

This is bread-and-butter engineering and physics: locating the centroid of a cross-section to find where a beam will bend, finding the center of gravity of a vehicle to judge whether it will tip, placing the balance point of a robot arm or an aircraft. Symmetry is your friend — if a body has a plane or axis of symmetry and its density respects that symmetry, the center of mass must lie on it, which often kills one or two of the integrals before you start. The common misconception worth dispelling: the center of mass need not lie inside the body at all (think of a doughnut or a boomerang), and it is not the same as the center of the bounding box.