About Cumulative Integration of data sets If you have a data set that can be modeled with a function or can be decently splined, you should integrate over the model or the spline. This approach is not a better candidate than the "Σ finite differences". For the given example both methods are exactly identical 2.7468 = 2.7468 [last values of each method]. The reality is more subtile: assume you have few irregular data, vg: the montly Thermies [Thermal Units], you have only 12 pairs of data that don't look much alike over the years. Splining introduces one more degree of incertitude. Unlike integration over the splined data set, the Σ finite diff's is closer to reality. On other hand, modeling a data set is such an art that most likely the model is de facto erroneous . Arguing too much about what's best, ends arguing that oval eggs might be splined square eggs.

define data here => and paste =>

About finite Diff's derivative of data sets You may want to try the centered recursive [Hildebrand] but you will be greatly deceived, though the recursive !may! show a slight smoothing effect on lightly noisy data sets. Recursive or this one proposed amplify noise or simply not enough decimal places ... try this data set with 3 decimal places to make your own proof.

These two tools will be most precious if you have: interpolated data [linterp, cinterp, ainterp]. These 3 aren't scalar in Smath, thus excluding further scalar processing down the WS.