Vertical chords of a hyperbola

I am starting with a generalization here, consider the equation of the hyperbola to be $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

We intend to find the average value of, say $f(x) = 2y = 2 \frac{b}{a} \sqrt{x^2 - a^2}$

The average value of $f(x)$ is denoted here as $\mu$

$\mu = \displaystyle \dfrac{\int_{a}^{2a} f(x) dx}{2a -a}$

$\int_{a}^{2a} f(x) dx = \int_{a}^{2a} 2\frac{b}{a} \sqrt{x^2 - a^2} dx = 2\frac{b}{a} [ \frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \ln (x + \sqrt{x^2 + a^2} )] \large{|_{a}^{2a}} = b[2\sqrt{3} - ln(2+\sqrt{3})]$

Substituting a = 3 and b = 2, we get

$\mu = 2[2\sqrt{3} - \ln(2+\sqrt{3}) ]$