I am given $\int_{0}^{2} \int_{-\sqrt{y^2 + 1}}^{\sqrt{y^2 + 1}} \int_{0}^{\sqrt{1 -x^2 + y^2}} dzdxdy$, I think the upper portion of the first limit is a hyperboloid of one sheet but isnt the solid the starting shape which would just be 1 ? how would i sketch this ?
2026-05-04 20:57:24.1777928244
Sketch the solid of integration Q.
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Notice that $z$ varies from $0$ to $\sqrt{1-x^2+y^2}$. When $z$ is equal to this upper limit, by squaring both sides we have: $$ z^2=1-x^2+y^2 $$ that is $$ y^2=z^2+x^2-1 $$ which is a hyperboloid of one sheet with its principal axis being the $y$ axis (the one that makes the hyperboloid a rotation surface over it)