Approximate an Integration by a linear formula

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 $f( x,y,w_1,\dots,w_n )$ $= e^{ - \frac{x^2 + y^2}{2}} \ln \left( \sum\limits_{i = 1}^n w_i e^{-\frac{(x-i)^2+(y-i)^2}{2} } \right)$.

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The TeX code in your question was abominable. Things like {{ \left( {x}^{2} \right) }} where (x^2) would suffice. Putting curly braces around long expressions where they serve no purpose. Lots of stuff like that. I've cleaned it up. – Michael Hardy Jul 6 '13 at 23:35
Thanks! I used MathType because I'm not good at Latex. – widapol Jul 7 '13 at 0:44

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 $w\to \bar w$ $$g(w) \approx g(\bar w) + \sum_i g_{w_i}(\bar w) (w-\bar w)_i$$
What is $f_{w_i} (x,y,\bar w)$ ? – widapol Jul 7 '13 at 0:49
the partial derivative of $f$ with respect to $w_i$ – Emanuele Paolini Jul 7 '13 at 0:58