Confusion with the definition of mean value For some reason the formula for mean started to trouble me: 
$$\mu  = \frac{1}{b-a}\int_a^b f(x)\:dx$$
The reason this confuses me a bit is because when I read this formula I read it as: $\text{Mean} = \frac{\text{Area}}{\text{Length}}$. I've used to the idea that mean is the average value of a set of numbers e.g. $\frac{1+2+3+4+5}{5} = 3$. Should I interpret this value as the average area under a curve or as the average value of a set function values? Picture will point out my question:

Hope I made my question clear :) Does mean value always equal $\text{Mean} = \frac{\text{Area (or volume)}}{\text{Length}}$. The definition confuses me because the integral doesn't equal the sum of function values $f(x)$, it is the area under the curve. This might be a very simple question but nonetheless confused me x) 

You can also think of the mean value of $f$ on $[a, b]$ as the average height of $f$ on the interval $[a, b]$. That is, $\mu$ is the unique constant which (when interpreted as a constant function) has the same integral as $f$ over $[a, b]$. Just like the mean of $k$ numbers is the unique number which (when interpreted as a collection of $k$ numbers) has the same sum as the sum of the $k$ numbers.
