<s>[INST] Volume between cone and sphere - First octant Find the volume between $z=\sqrt{x^2+y^2}$ and the sphere $x^2+y^2+z^2=1$ that lies in the first octant using cylindrical coordinates.
So I found the intersection and got $r=\frac{\sqrt2}{2}$.
I know theta has to be between $0$ and $\frac{\pi}{2}$ but not sure about z
 [/INST] Using Cylindrical coordinates:
$\displaystyle\int_{0}^{\frac{\pi}{2}}\int_{0}^{\frac{1}{\sqrt{2}}}\int_{r}^{\sqrt{1-r^2}}rdzdrd\theta=\int_{0}^{\frac{\pi}{2}}\int_{0}^{\frac{1}{\sqrt{2}}}{r(\sqrt{1-r^2}-r)}drd\theta=\int_{0}^{\frac{\pi}{2}}[-\frac{1}{3}(1-r^2)^{\frac{3}{2}}-\frac{r^3}{3}]_{0}^{\frac{1}{\sqrt{2}}}=\frac{\pi}{2}[(-\frac{1}{3(2\sqrt{2})}-\frac{1}{6\sqrt{2}})-(\frac{1}{3})]=\frac{\pi}{2}[\frac{-2\sqrt{2}}{12}+\frac{1}{3}]=\frac{\pi(2-\sqrt{2})}{12}$
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