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.

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, an approximate value of the double integral.

Finally, you can use the SimplicialCubature package after noticing that the region of integration can be split into two triangles (= simplicies). Moreover, the integrand is a polynomial function, and then SimplicialCubature offers the possibility to get the exact value of the integral.

library(SimplicialCubature)

S1 <- cbind(c(0,0), c(3,3), c(0,2)) # first triangle
S2 <- cbind(c(0,2), c(3,3), c(3,5)) # second triangle
S <- array(c(S1, S2), dim = c(2, 3, 2))

P <- definePoly(coef = c(1,1), k = cbind(c(1,0), c(0,1)))
printPoly(P) # x + y

integrateSimplexPolynomial(P, S)
# 24

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)

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.