Average value of a function over some given interval

Question

I was learning about simple harmonic motion from YouTube and in one of the lectures the teacher introduced to the formula for the average value of a function I know that formula for the average value of a function $f(x)$ over an interval $[a,b]$ is$$\frac{\displaystyle\int_{a}^{b}f(x)dx}{b-a}$$but I wanted to know the intuition behind that formula like how do we get to that formula. I searched for it in YouTube and Google I saw that formula everywhere but didn't find any explanation for it if someone has a explanation for it please share or refer some source from where I can know about it Thank You.

Answer 1

Suppose you plot the function $f(x)$ in the $[a,b]$ interval. Then we define the average value $\bar f$ in the following way: draw a line parallel to the $x$ axis at some height $h$. Calculate the area of the function $f$ above and below $h$. When these are equal, you get the average value. In mathematical terms: $$\int_a^b(f(x)-\bar f)dx=0\\\int_a^bf(x)dx=\bar f\int_a^bdx\\\bar f=\frac{\int_a^b f(x)dx}{b-a}$$

Answer 2

First think of a discrete average. Say you have $n$ people, each with some money. The average is the amount they each would have if they distributed all their money equally among all of them.

So the idea of an average is: same total amount of stuff, but distributed equally.

In the continuous case, the total amount of stuff is represented by area under the curve (integral). To distribute the stuff equally (so that the total amount remains the same), we want a constant function so that the total area under the constant function is equal to the total amount of area under the original curve. So geometrically we want a rectangule over the relevant interval, $[a,b]$, such that the rectangular area is equal to the area under the original function. If we call the height of the desired rectangle $h$, we want $$h\cdot (b-a)=\int_a^b f(x) dx$$ This height gives the average value of the function.

[Note: In general the function could have negative values, and/or the average value could be negative; so if you want you can think of signed area--(above the $x$-axis counts positive; below counts negative).]