Answer
a) $f_{ave}=\frac{4}{\pi}$
b) $c_1\approx 1.24$
$c_2\approx 2.81$
c) $[0,\pi]\times [0,4/\pi]$
Work Step by Step
a) Determine the average value on $[0,\pi]$:
$f_{ave}=\frac{1}{\pi}\int_0^{\pi}(2\sin x-\sin 2x)dx$
$=\frac{1}{\pi}\left((-2\cos x)_0^{\pi}-\left(-\frac{1}{2}\cos 2x\right)_0^\pi\right)$
$=\frac{4}{\pi}$
So $f_{ave}=\frac{4}{\pi}$.
b) Find $c$ so that $f(c)=f_{ave}$:
$2\sin c-\sin 2c=\frac{4}{\pi}$
$2\sin c-2\sin c\cos c=\frac{4}{\pi}$
$\sin c(1-\cos c)=\frac{4}{\pi}$
Drawing each side we find:
$c_1\approx 1.24$
$c_2\approx 2.81$
c) Find a rectangle with same area
$[0,\pi]\times [0,4/\pi]$
