Integration of two lines

First define a function from those points:

    fx <- approxfun(x, f_x)
    fy <- approxfun(y, f_y)
    f  <- function(x) abs(fx(x)-fy(x))

and now you can apply integrate() or trapz():

    xx <- sort(c(x, y))
    yy <- f(xx)
    trapz(xx, yy)

trapz() should return the more accurate (i.e. exact) result.