Question

Find the average value of the function σ(x, y) = x + y + x2 + y2 over the disk 0 ≤ x2 + y2 ≤ 4.

Answer 1

The average value of $\sigma(x,y)$ over the disk (call it $D$) is given by

$\dfrac{\displaystyle\iint_D\sigma(x,y)\,\mathrm dA}{\iint_D\mathrm dA}$

The denominator is just the area of $D$, which we know to be

$\displaystyle\iint_D\mathrm dA=\pi(2)^2=4\pi$

To compute the denominator, convert to polar coordinates, setting

$\begin{cases}x=r\cos\theta\\y=r\sin\theta\end{cases}\implies\mathrm dA=\mathrm dx\,\mathrm dy=r\,\mathrm dr\,\mathrm d\theta$

Then

$\displaystyle\iint_D\sigma(x,y)\,\mathrm dA=\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=2}(r\cos\theta+r\sin\theta+r^2)r\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle\int_{\theta=0}^{2\pi}\left(4+\dfrac83(\cos\theta+\sin\theta)\right)\,\mathrm d\theta=8\pi$

So the average value of $\sigma(x,y)$ over the disk is $\dfrac{8\pi}{4\pi}=2$.