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Core Concept Map Language

Theorems from Regions and order of integration

All theorems from Regions and order of integration.

Back to Regions and order of integration

[THEOREM]

Vertical-slice regions

If $c(x)\le d(x)$ on $[a,b]$ and the boundary functions are sufficiently regular for the integral to exist and if

\Omega=\{(x,y):a\le x\le b,\ c(x)\le y\le d(x)\},

then

\iint_\Omega f(x,y)\,dA =\int_a^b\int_{c(x)}^{d(x)} f(x,y)\,dy\,dx.

For each fixed $x$ , the inner variable $y$ runs from the lower boundary $c(x)$ to the upper boundary $d(x)$ .

[1]:Zero extension to a rectangle
[2]:Set up an integral over a planar region

[THEOREM]

Horizontal-slice regions

If $a(y)\le b(y)$ on $[c,d]$ and the boundary functions are sufficiently regular for the integral to exist and if

\Omega=\{(x,y):c\le y\le d,\ a(y)\le x\le b(y)\},

then

\iint_\Omega f(x,y)\,dA =\int_c^d\int_{a(y)}^{b(y)} f(x,y)\,dx\,dy.

For each fixed $y$ , the inner variable $x$ runs from the left boundary $a(y)$ to the right boundary $b(y)$ .

[1]:Zero extension to a rectangle
[2]:Set up an integral over a planar region

This page builds on Double integrals and Riemann sums.

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Vertical-slice regions Horizontal-slice regions