I understand that integrals are analogous to summing for continuous values, but for some reason I am still terrible at modeling continuous phenomena. Take for example calculating the diffraction of a wave traveling through the apeture shown below.
The book that I am following states that the amplitutde of a spherical wave moving away from the origin is
$$ u(x,y) = A \frac{e^{i (k r- \omega t)}}{r} $$
therefore, using Huygens’ principle with a plane wave incident normally on the aperture shown in the figure, the optical disturbance at $P_2$ is the sum of spherical waves emanating from the aperture:
$$ u(x,y) = A e^{-i \omega t} \iint_{a} \frac{e^{i k r_{12}}}{r_{12}} dx_a dy_a$$
where the integral is over the apeture area.
Now imagine that we discretized the problem so that the electric field at a point in the imaging plane is given by the sum of many spherical sources in the apeture plane:
$$ A e^{-i \omega t} \sum_i^M \sum_j^N \frac{e^{i k \sqrt{(x - x_{a_i})^2 + (y - y_{a_j})^2 + z^2} }}{\sqrt{(x - x_{a_i})^2 + (y - y_{a_j})^2 + z^2}} $$
How do we smooth out this sum? Typically we would have something of the form $\sum f(x_i) \Delta x \rightarrow \int f(x) dx$, so how do the $dx_a$ and $dy_a$ appear?