Problem 11 from chapter 16, section 9 ofCalculus with Analytic Geometryby Purcell and Varberg, reads as follows:

Evaluate $\displaystyle \iiint_{S}z^2 dV$, where $\displaystyle S$ is the region bounded by $\displaystyle x^2 + z = 1$ and $\displaystyle y^2 + z = 1$ and the $\displaystyle xy$-plane.

I envisioned the region as a pyramidal paraboloid centered at the origin with a height of 1 and a 2x2 base. I set up the following integral:

$\displaystyle 4\int_0^1\int_0^\sqrt{1 - z}\int_0^\sqrt{1 - z}z^2\,dx\,dy\,dz$

This gave me a result of $\displaystyle \frac{1}{3}$, which WolframAlpha tells me is correct: https://www.wolframalpha.com/input/?...t(1+-+z)%7D%5D

Along the way I had $\displaystyle 4\int_0^1z^2(\sqrt{1-z})^2dz$, which is how I'd have set it up if I knew only single integrals.

However, the answer key says $\displaystyle 0.8857$. I now wonder whether I interpreted the problem properly.