Why is the sum of the area under the density curve always greater than 1 (R)?

Question

I found codes for calculating the sum of the area under the density curve in R. Unfortunately, I do not understand why there is always an additional "0.000976" in the area ...

nb.data = 500000
y = rnorm(nb.data,10,2)

de = density(y)

require(zoo)
sum(diff(de$x[order(de$x)])*rollmean(de$y[order(de$x)],2))

[1] 1.000976

Why is this so?

It should be equal to 1, right?

Answer 1

- . , density, (.. trapzoidal rule), , , , . , :

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(.. ), . , (.. N) . , (, ).

, , , density . ( .)

, , - , 1, integrate :

> integrate(dnorm, lower=-Inf, upper=Inf, mean=10, sd=2)
1 with absolute error < 4.9e-06

Answer 2

. n ( - 512)

set.seed(42)
de = density(rnorm(500000, 10, 2))
sum(diff(sort(de$x)) * 0.5 * (de$y[-1] + head(de$y, -1)))
#[1] 1.00098

set.seed(42)
de = density(rnorm(500000, 10, 2), n = 1000)
sum(diff(sort(de$x)) * 0.5 * (de$y[-1] + head(de$y, -1)))
#[1] 1.000491

set.seed(42)
de = density(rnorm(500000, 10, 2), n = 10000)
sum(diff(sort(de$x)) * 0.5 * (de$y[-1] + head(de$y, -1)))
#[1] 1.000031

set.seed(42)
de = density(rnorm(500000, 10, 2), n = 100000)
sum(diff(sort(de$x)) * 0.5 * (de$y[-1] + head(de$y, -1)))
#[1] 1.000004

set.seed(42)
de = density(rnorm(500000, 10, 2), n = 1000000)
sum(diff(sort(de$x)) * 0.5 * (de$y[-1] + head(de$y, -1)))
#[1] 1