Triple integral in R and "Cubature" package in R and the manual solving the problem

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)
}
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,