Calculate the mean value of each of function below over one complete cycle of its graph.

1. $f(x) = \sin(x)$

   Hint (a):

   $y = \sin(x)$ has a period of $2π$.
   So choose $a$ and $b$-values such that $Δx = 2π$.

2. $g(x) =| \operatorname { sin } ( x ) |$

   Hint (b):

   Compare the periods of $y = |\sin(x)|$ and $y = \sin(x)$.

3. $k(x) = \sin^2(x)$

   Hint (c):

   Did you realize that since

   $\cos\Big(t-\frac{\pi}{2}\Big)=\sin(t), \sin^2(t)=\cos^2\Big(t-\frac{\pi}{2}\Big)=\cos^2(t)?$

   Steps (c):

   $\int_{0}^{2\pi}\sin^2(x)dx=\frac{1}{2}\int_{0}^{2\pi}\sin^2(x)+\cos^2(x)dx$

   Answer (c):

   $\frac{\int_{0}^{2\pi}g(x)dx}{2\pi - 0}=\frac{1}{2}$