Resolver por simulación la siguiente integral

\[\int_{0}^{2e} ln(x)dx = \lim_{b \to 0}\int_{b}^{2e} ln(x)dx\]

curve(log(x), from = 0, to = 2*exp(1))
abline(h = -2.9, col = 'goldenrod1')
abline(h = 1.7, col = 'goldenrod1')
abline(v = 0, col = 'goldenrod1')
abline(v = 2*exp(1), col = 'goldenrod1')

puntos = 10000
 d = c(); dn = c(); x = c(); y =c()
for(p in 1:puntos){
 x[p] = runif(1, 0, 2*exp(1))
 y[p] = runif(1, -2.9, 1.7)
 d[p] = (y[p]< log(x[p]))
 dn[p] = ifelse(d[p] == T, 'Abajo', 'Arriba')
}
 plot(x, y, col = ifelse(dn == 'Abajo', 'orange','gray'), pch = 19, cex = 0.25)

puntos = table(dn)
(prop = puntos[1.7]/sum(puntos)) # proporción de puntos en el area
##  Abajo 
## 0.7855
names(prop) = 'Area'
(area_total = 2*exp(1))
## [1] 5.436564
(area.simulada = area_total*prop)
##     Area 
## 4.270421