The average value of [tex]f(x,y)[/tex] over the annulus [tex]R[/tex] is given by
[tex]\dfrac{\displaystyle\iint_Rf(x,y)\,\mathrm dx\,\mathrm dy}{\displaystyle\iint_R\mathrm dx\,\mathrm dy}[/tex]
Converting to polar coordinates, we have
[tex]\dfrac{\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=a}^{r=b}\frac{6r}{r^2}\,\mathrm dr\,\mathrm d\theta}{\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=a}^{r=b}r\,\mathrm dr\,\mathrm d\theta}[/tex]
[tex]=\dfrac{12\pi\ln\frac ba}{(b^2-a^2)\pi}[/tex]
[tex]=\dfrac{12}{b^2-a^2}\ln\dfrac ba[/tex]
