The average value of [tex]\sigma(x,y)[/tex] over the disk (call it [tex]D[/tex]) is given by
[tex]\dfrac{\displaystyle\iint_D\sigma(x,y)\,\mathrm dA}{\iint_D\mathrm dA}[/tex]
The denominator is just the area of [tex]D[/tex], which we know to be
[tex]\displaystyle\iint_D\mathrm dA=\pi(2)^2=4\pi[/tex]
To compute the denominator, convert to polar coordinates, setting
[tex]\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[/tex]
Then
[tex]\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[/tex]
[tex]=\displaystyle\int_{\theta=0}^{2\pi}\left(4+\dfrac83(\cos\theta+\sin\theta)\right)\,\mathrm d\theta=8\pi[/tex]
So the average value of [tex]\sigma(x,y)[/tex] over the disk is [tex]\dfrac{8\pi}{4\pi}=2[/tex].
