Average of Accumulative Physical Quantity

In general, a weighted average of accumulative physical quantity can be determined by integration if the physical quantity can be expressed as a function of the interested weighting and quantified in the form of  infinitesimal elements. Besides the centroid of a curve, the centroid of an area can also determined by integration also.

Centrod of an Area

For an flat houmogeneous plate with uniform thickness, the centroid of the plate depends on its shape and the centroid is reduced to

Centroid of a Triangular area

For a triangular area with base b and height h, the moment along x axis direction is zero about the median, imply mean x lies on the median. For mean y is

Centroid of a Semicircular Area

For a semicircular area, due to symmetry about y axis,the centroid lies on the y axis. Imply mean y is

Centroid of a Quarter-circular area

For a quarter-circular area, the centroid is symmetry about the angle bisector, the moments about x axis and y axis are the same. imply mean x equal to mean y and is equal to

Centroid of a circular arc area

For a circular arc area, the centroid is symmetry about the angle bisector, x axis. The moments about x axis is zero, imply mean y is zero. The determination of mean x can make use of the moment of a circular arc wire. Imply mean x is