Every now and then a student will ask me why the formula for the density of a Normal random variable includes the constant, π, or more correctly (2π)-½.
The answer is that this term ensures that the density function is "proper" - that is, the integral of the function over the full real line takes the value "1". The area under the density, or "total probability", is "1".
Some students are happy with this (partial) answer, but others want to see a proof. Fair enough!
However, there's a trick to proving that this integral (area) is "1" in value. Let's take a look at it.