# Numerical double integral

how do i numerically calculate a double integral?
as i understand simpsons 3/8 rule is the optimal method for a single integral, is it still true for double integrals?
if so, how do i extend the 3/8s rule to do a double integral?

SteamKing
Staff Emeritus
Homework Helper
how do i numerically calculate a double integral?
as i understand simpsons 3/8 rule is the optimal method for a single integral, is it still true for double integrals?
if so, how do i extend the 3/8s rule to do a double integral?
I don't know what you mean by the Simpson's 3/8 rule being 'optimum'. It uses a different number of ordinates than Simpson's first rule (1-4-1), so that an even number of ordinates can be accommodated easily, where Simpson's first rule takes an odd number of ordinates.

Simpson's rules and the trapezoidal rule can be adapted to evaluate double integrals, and they tend to work well for regions which are triangular or rectangular. For more general regions, other methods may be easier to use for such evaluations; there are no hard and fast rules.

This paper illustrates a typical quadrature based on a 2-D Simpson's Rule:

https://www.math.ohiou.edu/courses/math3600/lecture24.pdf

Only with rectangles? so im guessing id have to be able to do some kind of change of variables then right?

SteamKing
Staff Emeritus
Homework Helper
Only with rectangles? so im guessing id have to be able to do some kind of change of variables then right?
Due to the nature of the Simpson's rule setup, a rectangular region will give the most accurate results with the least fuss, because the ordinates must be evenly spaced in a 2-D grid.

If the regions over which you are trying to evaluate a double integral are of a more general shape, perhaps a 2-D Gaussian quadrature method might give you fewer headaches about locating the ordinates in the region where the integrand must be evaluated.

what im doing in this case is i am doing a surface integral of a scalar function
using x(u,v),y(u,v),z(u,v) as the parameterized surface
so because this is of the form ∫∫something dudv where umin umax vmin vmax are all constants so its the same as a double integral over a rectangle and it should work right?