Mathematics

Problem

The function

g

is defined by

g(x)=\begin{cases}\sin x,&0\le x<\pi\\x-\pi,&\pi\le x\le 2\pi\end{cases}.

Determine the average value of

g

on the interval

\bigl[\tfrac{\pi}{2},\tfrac{3\pi}{2}\bigr]

. a)

\dfrac{1}{\pi}

\dfrac{1}{\pi}+\dfrac{\pi}{8}

\dfrac{1}{2\pi}+\dfrac{\pi}{16}

-\dfrac{1}{\pi}+\dfrac{\pi}{8}