Calculus II

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Integrable function

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Calculus II

Definition

An integrable function is a function for which the definite integral over a given interval exists and is finite. It must satisfy certain conditions, such as being bounded and having a limited number of discontinuities.

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5 Must Know Facts For Your Next Test

  1. A function is integrable on an interval $[a, b]$ if its absolute value has a finite integral over that interval.
  2. The Riemann Integrability criterion states that a bounded function on a closed interval $[a, b]$ is integrable if the set of its discontinuities has measure zero.
  3. Continuous functions on closed intervals are always integrable.
  4. If a function is monotonic (either non-increasing or non-decreasing), it is also integrable.
  5. Piecewise continuous functions with a finite number of discontinuities in $[a, b]$ are also integrable.

Review Questions

  • What conditions must be met for a function to be considered integrable?
  • How does the Riemann Integrability criterion determine if a function is integrable?
  • Are all continuous functions on closed intervals necessarily integrable?
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