Riemann sums — defining the integral

About this app

The definite integral as a limit of Riemann sums: a 3D column landscape under the curve, five example functions, choice of sample points (lower/upper/left/midpoint/right sum), a "refine" animation (columns grow ever thinner). Two singularity special cases show: for 1/√x the integral exists despite the pole, for 1/x² the sum grows beyond all bounds — the limit does not exist.

Subject: Analysis (one-dimensional)

More from Analysis (one-dimensional)

• Derivative as a sequence of secants
• Bounded growth
• Exponential growth and decay
• Optimisation: the largest rectangle
• Area between curves
• Area function is an antiderivative
• Sequential continuity
• Rational functions and asymptotes

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