Topics: Path Integral - Calculus

Path integrals can be used for many applications. Apart from the obvious use for calculating the area of a surface, there are many others such as calculating the average of a function over a curve or calculating the centre of mass given a density function.

Average of a Function

Let be a function and let be a curve in parametrised by .

The average of over , denoted is given by:

Note that the denominator is just the arc length of .

Centre of Mass

Let be a function that defines the density of an object in any one of its points. Let be a curve that defines the points where the object exists.

The total mass of the object defined by is given by:

The coordinates of its centre of mass are given by: