Type a command or search the notes. Tab cycles completions. Up and down recall history when blank. Escape dismisses, then leaves the prompt.
MATH
[1251]

Definitions from Regions and order of integration

All definitions from Regions and order of integration.

Back to Regions and order of integration

Zero extension to a rectangle

Let Ω\Omega be a bounded region contained in a rectangle RR. Define

fΩ(x,y)={f(x,y),(x,y)∈Ω,0,(x,y)∈R∖Ω.f_\Omega(x,y)= \begin{cases} f(x,y),&(x,y)\in\Omega,\\ 0,&(x,y)\in R\setminus\Omega. \end{cases}

If fΩf_\Omega is integrable on RR (for example, if it is bounded and piecewise continuous with discontinuities only along finitely many piecewise-smooth boundary curves), define the double integral over Ω\Omega by

∬Ωf(x,y) dA:=∬RfΩ(x,y) dA.\iint_\Omega f(x,y)\,dA :=\iint_R f_\Omega(x,y)\,dA.

The zero extension contributes nothing outside Ω\Omega, so the inner limits describe only the part of each slice that lies inside Ω\Omega.

  1. [1]:Double integral over a rectangle
  2. [2]:Vertical-slice regions
  3. [3]:Horizontal-slice regions
  4. [4]:Properties of double integrals

On this page

Zero extension to a rectangle