The integral2 function of the pracma package is a possibility:
library(pracma)
f <- function(x ,y) x + y
integral2(f, xmin = 0, xmax = 3, ymin = function(x) x, ymax = function(x) x+2)
If you are not allowed to use a package, you can nest the integrate function:
inner <- function(x) integrate(function(y) x+y, lower = x, upper = x + 2)$value
integrate(Vectorize(inner), lower = 0, upper = 3)
Both methods give 24.