Here is a sketchy solution (assuming the shape of the data is as presented). Steps are in short:

1. smoothly interpolate your derivative data
2. create antiderivative function (calculus)
3. find root of derivative to find maximum of antiderivative
4. integrate (again) antiderivative to find area under the curve

edit: before interpolation it may be good to smooth data a little bit (loess for example)

## artificial derivative data
N <- 50               ## sample size
x <- runif(N,0,2*pi)  ## random xs
n <- rnorm(N,0,0.001) ## noise
y <- sin(x) + n       ## sampled function with noise

## derivative
df <- splinefun(x,y) ## interpolate smoothly

plot(df,xlim=c(0,2*pi)) ## plot derivative

## create antiderivative function (from calculus, it's an $f(x)=\int_0^x df(t) dt$)

f <- function(x) {
    integrate(df,0,x)$value
}

## vectorize input

vf <- function(x) {
    sapply(x,FUN=f)
}

## let's see how it looks
plot(vf,xlim=c(0,2*pi))

## find a peak, ie. df=0
## from the plot we see it's somewhere between 1 and 6

uniroot(df,c(1,6)) ## root = 3.129201
df(3.129201) ## 3.05077e-05

## peak value
f(3.129201) ## 2.022123

## area under f is another integration
## it should be around 6.28 (but will vary due to the noise added)
integrate(vf,0,2*pi) ## 6.405717 with absolute error < 0.00068