Area under two curves in [R]

As it's Talk Like a Pirate Day, it seemed a good time to share a recent insight from working with the statistical system R. Why the pirate link? R, Jimlad...

Anyway, I recently had two overlapping curves, like in the first diagram, and wanted a way of finding the area that fell under the two without getting too complex, as one must normally do. I've a horrible feeling I'm about to say something really obvious as though it's profound (not for the first time), but having meddled for ages trying to find a way of doing it, it was something of a revelation to me suddenly to realize that for practical purposes (i.e., for simulations and graphs) you can find this area simply by taking the minimum y-value for each value of x. The lowest curve at any point is the upper-bound on the common area, so working across the x values taking the minimum is a really simple way to extract the shared region. Let's first create some sample data - these are the two curves in the diagram...

x <- seq(-3,3,0.1) #create a sequence of x values
y1 <- dnorm(x) #create the first curve
y2 <- dnorm(x + 0.75) # create the second curve
plot(x,y1) # graph the first curve
lines(x,y2,lty=2) # overlay the second

Then find the minimum at each point...

common <- array() #create a new array to hold the common area
for (i in 1:length(x)) { #for each value of x...
common[i] <- min(y1[i],y2[i]) #...take the lowest y across the two curves
}

This gives you the variable called 'common' which represents this shared area, and you can do all sorts of things with this, such as plotting a coloured region as I have in my second diagram.

polygon(c(min(x), x, max(x)), c(0, common, 0), col="lightblue")

As I say, this might be obvious to most of you, but it was a useful thing for me to work out so I hope it may help others. R, Jimlad.