Region

Boundary

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

Closed

The region contains all of its boundary points. As opposed to Open.

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.