Math 2.0

Real functions Definition

Improper integral

Builds on Riemann integralLimit at infinityOne-sided limit

Definition

Let f ⁣:[a,∞)→ℝf \colon [a, \infty) \to \mathbb{R} be Riemann integrable on [a,b][a, b] for each b>ab > a. The improper integral of ff over [a,∞)[a, \infty) converges if the following limit exists in ℝ\mathbb{R}: ∫a∞f(x) dx=lim⁡b→∞∫abf(x) dx.\int_a^\infty f(x) \, dx = \lim_{b \to \infty} \int_a^b f(x) \, dx. Similarly, let f ⁣:(a,b]→ℝf \colon (a, b] \to \mathbb{R} be Riemann integrable on [c,b][c, b] for each c∈(a,b)c \in (a, b). Then ∫abf(x) dx=lim⁡c→a+∫cbf(x) dx\int_a^b f(x) \, dx = \lim_{c \to a^+} \int_c^b f(x) \, dx, if this one-sided limit exists.

Remarks

If the limit does not exist, the improper integral diverges. For an integral over (−∞,∞)(-\infty, \infty), or with several bad points, we split the interval. Each piece must converge separately. For example, ∫1∞dxx2=1\int_1^\infty \frac{dx}{x^2} = 1, and ∫01dxx=2\int_0^1 \frac{dx}{\sqrt{x}} = 2. But ∫1∞dxx\int_1^\infty \frac{dx}{x} diverges.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Limit point of a set, definition, no evidence levelLimit point of a setdefinitionPartition of an interval, definition, no evidence levelPartition of an inter…definitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionLimit at infinity, definition, no evidence levelLimit at infinitydefinitionOne-sided limit, definition, no evidence levelOne-sided limitdefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionRiemann integral, definition, no evidence levelRiemann integraldefinitionImproper integral, definition, no evidence levelImproper integraldefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Limit point of a set definition
  2. Partition of an interval definition
  3. Limit of a function definition
  4. Mesh of a partition definition
  5. Riemann sum definition
  6. Limit at infinity definition
  7. One-sided limit definition
  8. Riemann integrable function definition
  9. Riemann integral definition
  10. Improper integral definition

Where this leads

41 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Improper_integral, Convergence of the integral
About this page
Tag0VCTBJMEJF
URIm2://0VCTBJMEJF
Kinddefinition
WritingW1 stated
MSC26A42, 40A10
DefinesImproper integral
UsesRiemann integral, Limit at infinity, One-sided limit

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by

Results about this