Triple Integrals over Elementary Regions

A solid is a footprint plus a ceiling and floor

A worked volume: the tetrahedron

Mass with a varying density

Choosing the inner variable

For the tetrahedron under $x + y + z = 1$ above the first quadrant, what are the inner bounds in the z-innermost setup?

What volume does the nested integral over the tetrahedron produce?

A solid sits between $z = x^2 + y^2$ and $z = 2 - x^2 - y^2$. What is its footprint in the $xy$-plane?

For the tetrahedron under $x + y + z = 1$, give the inner pass result $\int_0^{1 - x - y} 1\, dz$ in the form 1-x-y.

A solid of volume $2$ has mass $3$ under some density. Give the mean density, as a decimal like 1.5.