I would like to compute the triple integral of a function of three variables $f(x,y,z)$in R. I am using the package Cubature, Base, SimplicialCubature and the function adaptIntegrate(), Integrate and adaptIntegrateSimplex(). The integrand is equal to 1 only in certain domain(x<y<z, 0 otherwise).
Followings are the different ways to answer this question but i didn't understand how to compute the answer 0.166666 manually. If any member knows the answer, may reply. Using Cubature package,

library(cubature)
lower <- rep(0,3)
upper <- rep(1,3)
# First implementation (modified)
fxyz <- function(w) {
x <- w[1]
y <- w[2]
z <- w[3]
as.numeric(x <= y)*as.numeric(y <= z)
}
adaptIntegrate(f=fxyz,lowerLimit=lower,upperLimit= upper,doChecking=TRUE, maxEval=2000000,absError=10e-5,tol=1e-5)
Using Base package in R,
f.xyz <- function(x, y, z) ifelse(x < y & y < z, 1, 0)
f.yz <- Vectorize(function(y, z) integrate(f.xyz, 0, 1, y=y, z=z)\$value,
vectorize.args="y")

f.z <- Vectorize(function(z) integrate(f.yz, 0, 1, z=z)\$ value,
vectorize.args="z")
integrate(f.z, 0, 1)
Using SimplicialCubature package in R,

library(SimplicialCubature)
f <- function(x) 1
S <- CanonicalSimplex(3)
> adaptIntegrateSimplex(function(x) 1, S)
The integral is 0.1666667 but i don't understand how is that computed? If any member knows may reply with correct answer.