The average height of the surface [tex]z(x,y)[/tex] is the same as the average value of [tex]z(x,y)[/tex], so essentially you would compute
[tex]\dfrac{\displaystyle\iint_{[0,2]^2}(x^2+y^2)\,\mathrm dA}{\displaystyle\iint_{[0,2]^2}\mathrm dA}[/tex]
i.e. find the volume of the space under the paraboloid over the square, and divide that by the area of the square. The value of the denominator is easily seen to be 4, so we have
[tex]\displaystyle\frac14\int_{x=0}^{x=2}\int_{y=0}^{y=2}(x^2+y^2)\,\mathrm dy\,\mathrm dx=\dfrac83[/tex]
