I have two functions in form:
$$f(x,y)=\sum_{i}^{N}A_ig(\sqrt{(x-x_i)^2+(y-y_i)^2}) $$
where $A_i, x_i, y_i$ are different known values for both functions. $N$ is number of points. $g(r)$ is a function, usually something like $r^3$ or $r^2 \log(r)$.
I need to calculate area (volume) determined by these two funtions on a limited region.
I should compute this kind of an integral $$V = \iint \hspace{-5pt} \int_{f_1}^{f_2} \, dz \, dA$$
but I am lost, how to to this efficiently. I thought about using polar coordinates, since I have quadric roots, but I have stucked in $g(r)$.
You can express your volume as $V = V_2 - V_1$, where $$V_1 = \sum_i^N A_{1i} \iint g\left(\sqrt{(x-x_{1i})^2 + (y-y_{1i})^2}\right)dxdy$$ and $$V_2 = \sum_i^N A_{2i} \iint g\left(\sqrt{(x-x_{2i})^2 + (y-y_{2i})^2}\right)dxdy$$ where $f_1(x,y) = \sum_i^N A_{1i} g\left(\sqrt{(x-x_{1i})^2 + (y-y_{1i})^2}\right)$ and $f_2(x,y) = \sum_i^N A_{2i} g\left(\sqrt{(x-x_{2i})^2 + (y-y_{2i})^2}\right)$.
So if you can determine the function $$h(u,v) = \iint g\left(\sqrt{(x-u)^2 + (y-v)^2}\right)dxdy$$ you can use that function to calculate $$V = \sum_i^N \left(A_{2i}h(x_{2i},y_{2i}) - A_{1i}h(x_{1i},y_{1i})\right)$$