Average From Integral

Theorem

Suppose $R\subset {\mathbb{R}}^n$ Let $f:R\to \mathbb{R}$ be an Integrable function. Then, the average of $f$ on $R$ is:

$$A=\frac{1}{Vo{l}_n(R)}{\int }_{R}fd{V}_n$$

Where:

• $Vo{l}_n$ is the $n$-dimensional volume of $R$
• $d{V}_n$ is the $n$-dimensional volume element