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?

Half-full 1

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.)

Half-full 2

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.

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