Mean of Function Values
Definition
The average value of a function between $[a,\ b]$ and $f(x)$ is equivalent to dividing the integral of the function over the interval by the length of the interval.
$$ \dfrac{1}{b-a}\int_{a}^bf(x)dx $$
Derivation
Let’s denote a partition of the interval $[a,\ b]$ as $P$.
$$ P=\left\{ x_{1},\ x_{2},\ \cdots ,\ x_{n} \right\} $$
In this case, $a=x_{1} < x_{2} < \cdots < x_{n}=b$ and the distance between each point is the same. Also, $\Delta x=x_{i+1}-x_{i}$. We seek to approximate the average value of $f(x)$ by dividing $f(x_{i})$’s sum by $n$.
$$ \dfrac{ f(x_{1}) + f(x_{2}) + \cdots +f(x_{n}) } {n} $$
This implies that as $n$ increases, it will become closer to the average of the function values. Multiplying both numerator and denominator by $\Delta x$ gives the following.
$$ \dfrac{\Big( f(x_{1}) + f(x_{2}) + \cdots +f(x_{n}) \Big)\Delta x} {n \Delta x} $$
Since $n\Delta x=b-a$, it follows that:
$$ \dfrac{\Big( f(x_{1}) + f(x_{2}) + \cdots +f(x_{n}) \Big)\Delta x} {b-a} $$
Taking the limit where $n \rightarrow \infty$ and $\Delta x \rightarrow 0$, the numerator becomes $\int_{a}^bf(x)dx$.
$$ \dfrac{1}{b-a}\int_{a}^bf(x)dx $$
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Example
The average of one period of a trigonometric function is $0$.
Cosine Function
The average over one period of $\cos (kx)$ is as follows: $$ \int_{0}^\frac{2\pi}{k} \cos(kx)dx = \dfrac{1}{k}\left[ \sin(kx)\right]_{0}^{\frac{2\pi}{k}} =\dfrac{1}{k}(\sin 2\pi -\sin 0 ) =0 $$
Sine Function
The average over one period of $\sin (kx)$ is as follows: $$ \int_{0}^\frac{2\pi}{k} \sin(kx)dx = \dfrac{-1}{k}\left[ \cos(kx)\right]_{0}^{\frac{2\pi}{k}} =\dfrac{-1}{k}(\cos 2\pi -\cos 0 ) =0 $$