R, double integral with functional boundaries

Question

Is there an R function that handles double integrals where the region of the inner integration is a function of the outer variable? That is to say, while f(x) is integrated from -5 to 5, g(y) is integrated over 0 to h(x).

Thanks, JD

Answer 1

See the function integral2 in the pracma package which does exactly what you are asking for. Here is one of the examples from the help page:

##  Compute the volume of a sphere
f <- function(x, y) sqrt(1 -x^2 - y^2)
xmin <- 0; xmax <- 1
ymin <- 0; ymax <- function(x) sqrt(1 - x^2)
I <- integral2(f, xmin, xmax, ymin, ymax)
I$Q                             # 0.5236076 - pi/6 => 8.800354e-06

where one of the limits, ymax, is a function while the other limits are constants, but could also be functions.

Answer 2

You can use the adaptIntegrate function from the cubature library. I will assume for the sake of an example that your function is f(x, y) = x + y:

library(cubature)
fun <- function(x) x[1] + x[2]
adaptIntegrate(fun, c(0, 10), c(-5, 5), tol=1e-8)

This integrates over y from 0 to 10, then over x from -5 to 5.

Note that adaptIntegrate does not appear to accept a function as integral bounds. So if you want to the outer integral to go from 0 to h(x) you will probably have to write your own quadrature (not a hard thing to do).