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

The average value of over the disk (call it ) is given by



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



To compute the denominator, convert to polar coordinates, setting



Then




So the average value of over the disk is .