general solid region (Type I/II/III)

A solid squeezed between two surfaces over a shadow region in one of the coordinate planes.

A Type I solid is squeezed vertically between two surfaces $z=u_1(x,y)$ and $z=u_2(x,y)$ over a shadow $D$ in the $xy$ -plane — the solid's projection straight down onto that plane. Type II and Type III relabel which axis plays the role of "squeezed between two surfaces": a shadow in the $yz$ -plane bounded in $x$ , or a shadow in the $xz$ -plane bounded in $y$ . Choosing the right type is entirely about which direction gives the simplest inner bounds.

Examples

For $E$ bounded above by $z=x+y$ , below by $z=0$ , over the triangle $D$ with vertices $(0,0)$ , $(1,0)$ , $(0,1)$ : $\iiint_E xyz\,dV=\int_0^1\!\int_0^{1-x}\!\int_0^{x+y}xyz\,dz\,dy\,dx$ .