Question

Find the average value of the function f(x, y) = 7/ x2 + y2 on the annular region a2 ≤ x2 + y2 ≤ b2, where 0 < a <
b.

Answer 1

Denote the annular region by $A$. The average value of $f(x,y)=\dfrac7{x^2+y^2}$ over $A$ is given by

$\dfrac{\displaystyle\iint_A f(x,y)\,\mathrm dA}{\displaystyle\iint\mathrm dA}$

For both integrals, convert the region to polar coordinates. We have

$\displaystyle\iint_Af(x,y)\,\mathrm dA=\int_{\theta=0}^{\theta=2\pi}\int_{r=a}^{r=b}\frac7{r^2}r\,\mathrm dr\,\mathrm d\theta$
$=14\pi\displaystyle\int_{r=a}^{r=b}\frac{\mathrm dr}r$
$=14\pi\ln\dfrac ba$

$\displaystyle\iint_A\mathrm dA=\int_{\theta=0}^{\theta=2\pi}\int_{r=a}^{r=b}r\,\mathrm dr\,\mathrm d\theta$
$=\pi(b^2-a^2)$

and so the average value is

$\dfrac{14}{b^2-a^2}\ln\dfrac ba$