Average Value

Formula: f_avg = (1/(b-a)) · ∫ from a to b f(x)dx

In English: The average height of the function over [a,b].

Example: Average value of f(x) = x² on [0, 3]:

f_avg = (1/3) · ∫ from 0 to 3 x² dx = (1/3) · [x³/3] from 0 to 3 = (1/3)(9) = 3

Mean Value Theorem for Integrals:

There exists c in [a,b] where f(c) = f_avg. The function equals its average at some point.

AP exam: "Find the average velocity" = average value of velocity function. "Find the average temperature" = average value of temperature function. Same formula every time.

Reference:

Wikipedia: Average Value

https://en.wikipedia.org/wiki/Mean_of_a_function