Calc III · §15.6 triple integrals over general regions

Triple integrals: Type 1, 2, 3 & the 2D shadow

A solid E can be sliced three ways. Each type integrates one variable first (between two surfaces), then projects the leftover onto a coordinate plane — where the shadow D is itself a Type I or Type II plane region. Pick a type, watch the ray sweep into a slab into the solid, and read off the full iterated integral.

Build the integral

Solid E

Solid type — integrate this variable first

The shadow D — slice it as

Sweep the slab across D

Show the solid E shadow D on the plane current slab (middle integral) sample ray (inner integral) all strips on D (2D)

Ray → slab → solid: the inner integral is a ray between the two surfaces, the middle integral sweeps it along a strip, the outer integral sweeps strips across D.

The solid, its shadow, and the sweeping slab

The shadow D as a plane region

The iterated integral

Same solid, three types — each is a different order of the same ∭. 🧊