Answer
$$
f(x)=(2 x-1)^{1 / 2}
$$
The average value of a function $f(x)$ on the interval$[1,13]$ is given by:
$$
\begin{aligned}
\text{The average value}&=\frac{1}{b-a} \int_{a}^{b}(f(x)) d x\\
&=\frac{1}{13-1} \int_{1}^{13}(2 x-1)^{1 / 2} d x\\
&\approx 3.444
\end{aligned}
$$
Work Step by Step
$$
f(x)=(2 x-1)^{1 / 2}
$$
The average value of a function $f(x)$ on the interval$[1,13]$ is given by:
$$
\begin{aligned}
\text{The average value}&=\frac{1}{b-a} \int_{a}^{b}(f(x)) d x\\
&=\frac{1}{13-1} \int_{1}^{13}(2 x-1)^{1 / 2} d x\\
&=\frac{1}{12}\left(\frac{1}{2}\right) \int_{1}^{13} 2(2 x-1)^{1 / 2} d x\\
&=\left.\frac{1}{24} \cdot \frac{2}{3}(2 x-1)^{3 / 2}\right|_{1} ^{13}\\
&=\frac{1}{36}\left(25^{3 / 2}-1\right)\\
&=\frac{1}{36}(125)-\frac{1}{36}\\
&=\frac{124}{36} \\
&=\frac{31}{9} \\
&\approx 3.444
\end{aligned}
$$
