We have spent some time now looking at how we might numerically approximate the integrals of functions but have thus far completely ignored the problem of integrating functions with more than one argument, known as multivariate functions.

When solving integrals of multivariate functions mathematically we typically integrate over each argument in turn, treating the others as constants as we do so. At each step we remove one argument from the problem and so must eventually run out of them, at which point we will have solved it.

It is consequently extremely tempting to approximate multivariate integrals by recursively applying univariate numerical integration algorithms to each argument in turn.

When solving integrals of multivariate functions mathematically we typically integrate over each argument in turn, treating the others as constants as we do so. At each step we remove one argument from the problem and so must eventually run out of them, at which point we will have solved it.

It is consequently extremely tempting to approximate multivariate integrals by recursively applying univariate numerical integration algorithms to each argument in turn.

Full text...