Originally Posted by

**HallsofIvy** Unfortunately, the original method is wrong so pretty much everything everyone has said is off the point. The problem is NOT just simplifying that equation, the equation itself is wrong.

You do NOT find the average of a **function**, f(x) on an interval, [a, b], by $\displaystyle \frac{f(b)- f(a)}{b- a}$, any more than you find the average of a list of numbers by simply averaging the first and last values. you have to **sum** the numbers which, for functions, means "integrate".

The average of f(x) on the interval [a, b] is

$\displaystyle \frac{1}{b-a} \int_a^b f(x)dx$

The average you want is

$\displaystyle \frac{2g\int_{\pi/4}^{\pi/3} (1- cos(\theta))d\theta}{\frac{\pi}{3}- \frac{\pi}{4}}$