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Propositions from Regions and order of integration

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Properties of double integrals

For integrable ff and gg over Ω\Omega and a constant kk,

∬Ω(f+g) dA=∬Ωf dA+∬Ωg dA,\iint_\Omega(f+g)\,dA =\iint_\Omega f\,dA+\iint_\Omega g\,dA,∬Ωkf dA=k∬Ωf dA.\iint_\Omega kf\,dA =k\iint_\Omega f\,dA.

If Ω=Ω1∪Ω2\Omega=\Omega_1\cup\Omega_2 and the two regions are disjoint, then

∬Ωf dA=∬Ω1f dA+∬Ω2f dA.\iint_\Omega f\,dA =\iint_{\Omega_1}f\,dA+\iint_{\Omega_2}f\,dA.

Splitting the parabolic region above is an instance of additivity.

  1. [1]:Zero extension to a rectangle
  2. [2]:Area as a double integral

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Properties of double integrals