How do you find the average value of a function?

Category: Integrals »

Course: Calculus »

Relevant MM Lessons:

Overview: Average Function Value

This question has its own dedicated lesson ». Make sure to check it out if you need a more thorough read, or want to become an expert on how average function value shows up on tests.

Function Average Value

The average value of a function is both intuitive and odd. It is intuitive in that it represents the "average height" of the function over a certain interval. Since the "height" of a function at a specific $x$ location is the function's value at that $x$, it makes sense that the average function value can be visually interpreted as the average height of the function.What makes it odd is that we're talking about taking an average of something continuous, which goes against our raw intuition from Pre-Algebra era definitions of computing averages. We usually think of averages as "sum up all the numbers and divide by the number of terms". Fortunately, while we may prefer the more comfortable scenario of averaging a discrete number of items, the formula for average function value also follows the idea of summing all values and dividing.The average function value for a function $f(x)$ on the interval $[a,b]$ is given by:$$\mathrm{Avg} \,\,\,\, \mathrm{val} = \frac{\int_{a}^{b} f(x) \, dx }{b-a}$$Compare this with a discrete average of a list of $x$ values, ${x_1, \,\, x_2, \,\, \dots \,\, x_n}$:$$\mathrm{Avg} \,\,\,\, \mathrm{val} = \frac{\sum_{i=1}^{n} x_i}{n}$$Again, this is very similar conceptually, so hopefully that helps you remember this new integral formula if you have to memorize it!