Average value of a function

Averaging a finite list of numbers means adding them and dividing by how many there are. A continuous function takes infinitely many values across an interval, so the sum becomes an integral and the count becomes the interval's width. That gives the formula f_avg = (1 / (b − a)) ∫ from a to b of f(x) dx.

Geometrically, the average value is the height of a rectangle spanning [a, b] whose area equals the area under the curve. For f(x) = x² on [0, 3], the definite integral equals 9, so the average value is 9 / 3 = 3 — a rectangle 3 units wide and 3 units tall has the same area as the region beneath the parabola.

The Mean Value Theorem for Integrals guarantees that a continuous function actually attains its average value: there is at least one c in [a, b] with f(c) = f_avg. That is what makes the rectangle interpretation meaningful rather than merely arithmetic.

Be careful not to confuse average value with average rate of change, which is (f(b) − f(a)) / (b − a) and uses only the endpoints. A useful connection ties the two together: the average value of a velocity function over a time interval equals the object's average velocity, since displacement divided by elapsed time is exactly the integral of velocity divided by the interval width. AP Calculus AB tests this topic in its applications of integrals unit, often with a rate function given in context and a question asking for the average rate over a stated interval.