Math 2.0

Real functions Definition

Riemann integral

Builds on Riemann integrable function

Definition

Let a<ba < b and let f ⁣:[a,b]→ℝf \colon [a, b] \to \mathbb{R} be Riemann integrable. The Riemann integral of ff over [a,b][a, b] is the unique number II in the definition of Riemann integrability. It is written ∫abf(x) dx.\int_a^b f(x) \, dx.

Remarks

For f≥0f \ge 0, the integral equals the area under the graph of ff and above the xx-axis. By convention, ∫aaf=0\int_a^a f = 0 and ∫baf=−∫abf\int_b^a f = -\int_a^b f. The fundamental theorem of calculus computes the integral from an antiderivative. The integral is named after Bernhard Riemann.

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Partition of an interval, definition, no evidence levelPartition of an inter…definitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionRiemann integral, definition, no evidence levelRiemann integraldefinition
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  1. Partition of an interval definition
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Further reading

  1. Web https://en.wikipedia.org/wiki/Riemann_integral, Riemann integral
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