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MATH
[1251]

Theorems from Double integrals and Riemann sums

All theorems from Double integrals and Riemann sums.

Back to Double integrals and Riemann sums

Criterion for Riemann integrability

A bounded function ff is Riemann integrable over RR if and only if, for every ε>0\varepsilon>0, there is a partition PεP_\varepsilon such that

0≤U(f,Pε)−L(f,Pε)<ε.0\le U(f,P_\varepsilon)-L(f,P_\varepsilon)<\varepsilon.

In that case,

∬Rf dA=sup⁡PL(f,P)=inf⁡PU(f,P).\iint_R f\,dA =\sup_P L(f,P) =\inf_P U(f,P).

This is equivalent to requiring that the tagged sums have a common limit. It is usually the easier criterion for proving integrability or non-integrability.

  1. [1]:Upper and lower double Riemann sums
  2. [2]:A function that is not integrable
  3. [3]:Riemann integrability and an upper sum

Continuous functions are integrable

Every continuous function on a rectangle is Riemann integrable over that rectangle.

On sufficiently small rectangles, continuity prevents large variation in ff, so the upper and lower sums can be made arbitrarily close.

Continuity is sufficient, not necessary: a discontinuous function may still be integrable.

  1. [1]:Double integral over a rectangle

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Criterion for Riemann integrabilityContinuous functions are integrable