Average Value of a Function

Average Value of a Function

The average value of $f$ on $[a, b]$ is:

$f_{\text{avg}} = \frac{1}{b-a} \int_a^b f(x)\, dx$

This generalizes the discrete average $\frac{f_1 + f_2 + \cdots + f_n}{n}$ to a continuous function.

Mean Value Theorem for Integrals

If $f$ is continuous on $[a, b]$, there exists $c \in [a, b]$ such that:

$f(c) = f_{\text{avg}} = \frac{1}{b-a} \int_a^b f(x)\, dx$

In other words, $f$ actually achieves its average value somewhere in the interval.

Geometric Interpretation

The average value is the height of a rectangle with base $(b-a)$ that has the same area as the region under $f$:

$f_{\text{avg}} \cdot (b-a) = \int_a^b f(x)\, dx$

Examples

• $f(x) = x$ on $[0, 4]$: $f_{\text{avg}} = \frac{1}{4} \cdot 8 = 2$ (the midpoint — makes sense for a linear function)
• $f(x) = x^2$ on $[0, 3]$: $f_{\text{avg}} = \frac{1}{3} \cdot 9 = 3$
• $f(x) = 1$ on any interval: $f_{\text{avg}} = 1$ (average of a constant is itself)

Applications

• Average temperature over a day
• Average power consumed over a cycle
• Average speed over a journey (area under speed-time curve divided by time)

Your Task

Implement double average_value(double (*f)(double), double a, double b, int n) using the midpoint rule internally.