Numerical integration & differentiation

What is Numerical integration & differentiation?

When you have data points rather than a formula, calculus is done numerically. The trapezoidal rule approximates the area under the curve by summing trapezoids between adjacent points, giving a running (cumulative) integral. The central-difference formula estimates the local slope from each point's neighbors.

No model is fitted — both operate directly on the samples, so accuracy depends on how densely the curve is sampled. Differentiation amplifies noise, so smoothing the series first is often advisable before taking a derivative.

When should I use Numerical integration & differentiation?

• Total area / accumulated quantity under sampled data (e.g. area under a concentration curve).
• Local rate of change (slope) at each point without a fitted model.

What data does it need?

An X column and a Y column (numeric, ordered by X).

What does it report?

The cumulative integral and the point-wise derivative, plotted against X on a dual axis, plus the total integral.

What does it assume?

• X is sorted and reasonably dense; coarse sampling degrades both estimates.
• Differentiation magnifies noise — smooth first for noisy Y.

How do I interpret the result?

The final cumulative value is the total area under the curve. Derivative zero-crossings mark the peaks and troughs of Y.

See also

• Interpolation (spline / linear)
• Data smoothing (Savitzky-Golay / MA / LOWESS)
• Trapezoidal AUC (paired X-Y curve)

References

• Press et al. (2007). Numerical Recipes, 3rd ed., ch. 4 (integration) & 5 (differentiation).