To calculate the AUC of a curve, you can use a numeric integration function such as trapz()
in the caTools package.
auc <- trapz(recall, precision)
The F-score is the harmonic mean for a given cutoff value. In your case, you would get many F-scores for each curve so it would not summarize the curve as you like.
The AUC describes the performance of the model across possible values of the continuous output from the model. The F-score describes a model at a particular cutpoint. It is more of a way to combine recall and precision to a single statistic.
Be careful when explaining it though. Usually, AUC is discussed in the context of sensitivity and specificity.