# Half-full and Half-empty

A thought experiment:

Suppose you had a container with a rectangular base whose length runs from x = a to x = b, with a width of one-inch. The container has four vertical rectangular sides. You put a piece of, say, plastic into the container which fits snugly along the bottom and four sides. The top of the piece is irregular and has the equation y = f(x).  If the plastic were to melt, how high up the sides would the melted plastic rise?

One way to think about this is to consider the final level, L. When melted, the plastic above the final level must fill in the part below, leaving a rectangle with the same area as that under the original function’s levels. (The one-inch width will remain the same and not affect the outcome.)

So the original area is $\displaystyle \int_{a}^{b}{f\left( x \right)dx}$ and the final area is $L\left( b-a \right)$. Since these are the same we can write and equation and solve it for L.

$\displaystyle L\left( b-a \right)=\int_{a}^{b}{f\left( x \right)dx}$

$\displaystyle L=\frac{\int_{a}^{b}{f\left( x \right)dx}}{b-a}$

But that’s the equation for the average value of a function!

What a surprise!

Well, not a surprise for you, the teacher. This might be a good way to sneak up on the average value of a function idea for your students while giving them a good visual idea of the concept.