Math 2.0

Real functions Theorem

Additivity of the integral over intervals

Builds on Definite integral

Statement

Let a<c<ba < c < b and let f ⁣:[a,b]→ℝf \colon [a, b] \to \mathbb{R}. Then ff is Riemann integrable on [a,b][a, b] if and only if it is Riemann integrable on [a,c][a, c] and on [c,b][c, b]. In this case, ∫abf(x) dx=∫acf(x) dx+∫cbf(x) dx.\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx.

Remarks

With the sign convention of the definite integral, the formula holds for any order of aa, bb, cc. This needs ff to be integrable on the largest of the intervals. Mathlib states it as intervalIntegral.integral_add_adjacent_intervals.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
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 integraldefinitionDefinite integral, definition, no evidence levelDefinite integraldefinitionAdditivity of the integral over intervals, statement, no evidence levelAdditivity of the int…statement
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. Partition of an interval definition
  2. Mesh of a partition definition
  3. Riemann sum definition
  4. Riemann integrable function definition
  5. Riemann integral definition
  6. Definite integral definition
  7. Additivity of the integral over intervals statement

Where this leads

16 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Integral, Linearity
About this page
Tag05RCDKHZM6
URIm2://05RCDKHZM6
Kindstatement, theorem
WritingW0 stub
MSC26A42
StatusThis copy of the site has no build data. Run m2 status 05RCDKHZM6.
UsesDefinite integral

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (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