### Home > CALC > Chapter 6 > Lesson 6.4.2 > Problem6-141

6-141.

Find and compare the average value of each of these functions over one complete cycle.

For each of the functions below, we are given $f(x)$ and we want to find the average value of $f(x)$.

1. $f(t) =\operatorname{sin} t$

$y =\operatorname{sin}x$ has a period of $2π$.
So choose $a$ and $b$ values such that $Δx = 2π$.

1. $g ( t ) = | \operatorname { sin } t |$

Compare the periods of $y = |\operatorname{sin}x|$ and$y =\operatorname{sin}x$.

1. $k(t) = (\operatorname{sin}t)^2$

Did you realize that since

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

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