Volume of a given solid?

Question

Passed by a question now, need a volume of a given solid but I can't understand how to draw such a solid.

The region is given by : $S=\{(x,y,z)\in \mathbb{R}^3 : |x| + 2|y| \le 1-z^2\}.$

Any suggestion?

Answer 1

We compute the volumes as integrals of cross-sections.

Consider the sections given by the intersection of our solid $S$ and the plane $z=t$ with $t\in [-1,1]$ (because $|x| + 2|y|\geq 0$ implies $1-t^2\geq 0$).

Note that $|x| + 2|y|\leq 1-t^2$ is a rhombus in the plane $z=t$ with diagonals along the $x$-axis and $y$-axis. It is easy to see that the lengths of those diagonals are $2(1-t^2)$ and $(1-t^2)$. Therefore its area is $\frac{1}{2}[2(1-t^2)\cdot(1-t^2)]=(1-t^2)^2$. Therefore $$\mbox{volume}(S)=\int_{t=-1}^1\left(\iint_{|x| + 2|y|\leq 1-t^2}1 dx dy\right) dt\\=\int_{-1}^1\mbox{area}\left(\{|x| + 2|y|\leq 1-t^2\}\right) dt\\ =\int_{-1}^1 (1-t^2)^2 dt=\frac{16}{15}.$$