Region

Boundary

The subset of a region not belonging to the inerior of the region.

Closed

The region contains all of its boundary points.

Open

The region contains none of its boundary points.

Simply-Connected

One piece with no holes.

Bounded

The region has finite size.

Simple

2D

Vertically Simple

(Type 1 Region) A region bounded by the vertical lines $x=a$ and $x=b$, and the curves $y=g(x)$ and $y=h(x)$, where $g(x)<h(x)$ and $a<b$

$${\iint }_{R}f(x,y)dA={\int }_{x=a}^{x=b}{\int }_{y=g(x)}^{y=h(x)}f(x,y)dydx$$

Horizonally Simple

(Type 2 Region) A region bounded by the horizontal lines $y=c$ and $y=d$, and the curves $x=g(y)$ and $x=h(y)$, where $g(y)<h(y)$ and $c<d$

$${\iint }_{R}f(x,y)dA={\int }_{y=c}^{y=d}{\int }_{x=g(y)}^{x=h(y)}f(x,y)dxdy$$

3D

X-Simple

If the 2D region $D$ is the projection of 3D region $E$ onto the $zy$-plane and $E$ is exclusively bounded above and below by surfaces $x=u_2(y,z)$ and $x=u_1(y,z)$ then

$${\iiint }_{E}f(x,y,z)dV={\iint }_D\left({\int }_{x=u_1(y,z)}^{x=u_2(y,z)}f(x,y,z)dx\right)dA$$

and then $E$ is an $x$-simple region.

Y-Simple

If the 2D region $D$ is the projection of 3D region $E$ onto the $xz$-plane and $E$ is exclusively bounded above and below by surfaces $y=u_2(x,z)$ and $y=u_1(x,z)$ then

$${\iiint }_{E}f(x,y,z)dV={\iint }_D\left({\int }_{y=u_1(x,z)}^{y=u_2(x,z)}f(x,y,z)dy\right)dA$$

and then $E$ is a $y$-simple region.

Z-Simple

If the 2D region $D$ is the projection of 3D region $E$ onto the $xy$-plane and $E$ is exclusively bounded above and below by surfaces $z=u_2(x,y)$ and $z=u_1(x,y)$ then

$${\iiint }_{E}f(x,y,z)dV={\iint }_D\left({\int }_{z=u_1(x,y)}^{z=u_2(x,y)}f(x,y,z)dz\right)dA$$

and then $E$ is a $z$-simple region.