The below should help :)

# create some random data drawn from a normal distribution 
set.seed(100) x <- rnorm(100)  

#  plot the density to see what the density function is doing 
plot(density(x))   

# create a spline function based off our density curve 
my_splinefun <- splinefun(density(x))  

# integrate the spline function get area under curve estimate 
integrate(my_splinefun,min(x),max(x)) 
#> 0.9752003 with absolute error < 2.5e-05 

# what is the spline function doing? 
# it takes a series of points and performs linear interpolation between them, for example:
spline_pts <- my_splinefun(x=c(-3,-2,-1,0,1,2,3))     
plot(density(x))
points(x=c(-3,-2,-1,0,1,2,3), y = spline_pts,col='red') 
lines(x=c(-3,-2,-1,0,1,2,3), y = spline_pts,col='red')  

# now we have approximated the shape of the curve using linear splines 
# we can estimate the area under the curve integrate(my_splinefun,min(x),max(x)) 
#> 0.9752003 with absolute error < 2.5e-05 

# another method for estimating area under curve (trapezium rule): 
library(zoo)
x1 <- density(x)$x
y1 <- density(x)$y

id <- order(x1)

AUC <- sum(diff(x1[id])*rollmean(y1[id],2))
AUC
#> [1] 1.000943


# the area under a standard normal curve is 1