Image by Benjamin O. Tayo

Integrals By Symmetry

In so many problems in mathematics, physics, and engineering, we just want to know if the value of an integral is zero or non-zero

I. Introduction

There are so many problems in mathematics, physics, and engineering where we would like to know if the value of an integral is zero or non-zero. For example, determination of the electric field from different charge distributions such as line charge, surface charge, or volume charge; determination of the inertia tensor in mechanics; the calculation of matrix elements in perturbation theory in quantum mechanics; or the calculation of dipole transition probability for selection rules in atomic and molecular physics. In this article, we show how simple symmetry rules can be used to determine if an integral is going to vanish or not.

II. Examples

II.1 Sin(x) Function

Figure 1. Plot of sin(x), shows an asymmetric function. Image by Benjamin O. Tayo

II.2 Cos(x) Function