We can try a to approximate the area under the curve using Trapezoidal rule on the
plotting coordinates that density() produces.
nbin <- 1024 # number of bin
d <- density( rnorm(50000), n=nbin)
totalArea <- 0
for(i in 1:(nbin-1) ){
xxx <- d$x[i+1] - d$x[i] # width of bin
yyy <- (d$y[i+1] + d$y[i])/2 # average height of bin
binArea <- xxx*yyy
totalArea <- totalArea + binArea
}
print(totalArea)
We can see that the total area under the curve is close to 1 and the approximation
gets better as nbin is increased (but this is always an overestimate due to the
concavity of the normal curve).
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