**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 and the final area is . Since these are the same we can write and equation and solve it for *L.*

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.

### Like this:

Like Loading...