Approximate an Integration by a linear formula

Question

I just wonder are there any methods to approximate the following integration by a linear formula ? $$ \int_{x_1}^{x_2} \int_{y_1}^{y_2} f( x,y,w_1,\dots,w_n ) \, dx \, dy \approx \sum\limits_{i = 1}^n a_i w_i + b, $$ where $a_i$ and $b$ are constants. For example $f( x,y,w_1,\dots,w_n ) $ $= e^{ - \frac{x^2 + y^2}{2}} \ln \left( \sum\limits_{i = 1}^n w_i e^{-\frac{(x-i)^2+(y-i)^2}{2} } \right)$.

Answer 1

Let $$ g(w) = \iint f( x,y,{w_1},...,{w_n}) $$ then $$ g_{w_i}(\bar w) = \int\int f_{w_i} (x,y,\bar w) dx\,dy $$ and for $w\to \bar w$ $$ g(w) \approx g(\bar w) + \sum_i g_{w_i}(\bar w) (w-\bar w)_i $$