book recommendation numerical double integration

Question

I am looking for references (books, articles, or forums) that treat the approximation of a double numerical integration by a double sum and how to implement it on Matlab. Thank you.

Answer 1

No book recommendation (really, any text-book on numerical integration should cover it). In general, it is possible to iterate the rule for a single integral. For example, trapezoid:

$$\int_a^{a+h}f(x)dx\approx \frac{h}{2}(f(a)+f(a+h))$$

$$\int_b^{b+h}\int_a^{a+h} f(x,y)dxdy\approx \int_b^{b+h} \frac{h}{2}(f(a,y)+f(a+h,y))dy\approx \frac{h^2}{2^2}(f(a,b)+f(a+h,b)+f(a,b+h)+f(a+h,b+h))$$

It's also possible to integrate the whole area using Monte-Carlo. In this case for example, envelope the area into a multi-dimensional cube, and count the hits from an multidimensional uniform variable covering the cube. The area is proportional to the hits. Then take the average of the function values on the hits and scale by the area.