Consider the oval $x^2 + 2y^2 < N$, the number of integer points is approximately $2\pi N$. Now consider the average:
$$ \frac{1}{2\pi N} \sum_{\{ (x,y): x^2 + 2y^2 < N\}} (x + \sqrt{2}\, y\big) $$ This is the average over all integer points inside the ellipse. I am guessing this is close to the Riemann integeral: $$ \int_0^1 t \, dt = \frac{1}{2} $$ I believe it is necessary to restrict to the first quadrant, so that $x,y > 0$.
I plotted the function $f(N) = \#\{ (x,y) \in [0, \infty)^2 : x^2 + 2\, y^2 < N \}$ and we get a devil's staircase type function, approximating a straight line.
To approximate with a line we should say $f'(x) \approx \frac{2}{\pi} $ yet we have that $f'(x) = 0$ everywhere except at the jump discontinuities.