 # I need to find the volume of the intersection set A &#x2229;<!-- ∩ --> B whereby A Aganippe76 2022-07-07 Answered
I need to find the volume of the intersection set $A\cap B$ whereby $A=\left\{\left(x,y,z\right)\in \mathbb{R}|{x}^{2}+{y}^{2}+{z}^{2}\le 1\right\}$ and $B=\left\{\left(x,y,z\right)\in \mathbb{R}|{x}^{2}+{y}^{2}\le 1/2\right\}.$
It is clear that $A$ represents the unit ball centered at the origin and $B$ represents the cylinder with radius $\frac{1}{\sqrt{2}}.$. I should at some point use the Fubini theorem. I am puzzled by intersection set. Can somebody provide a solution proposal or a comment? Thanks.
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The intersection is what is both inside the cylinder and sphere, i.e. what is in the cylinder and below/above the caps it cuts out from the sphere.
Using the reflection symmetry in $z$ about 0, the integral is given by
$2\int {\int }_{{x}^{2}+{y}^{2}\le 1/2}\sqrt{1-{x}^{2}-{y}^{2}}\mathrm{d}A=2\int {\int }_{{x}^{2}+{y}^{2}\le 1/2}\sqrt{1-{x}^{2}-{y}^{2}}\mathrm{d}x\mathrm{d}y$
And in polar coordinates
$4\pi {\int }_{0}^{1/\sqrt{2}}r\sqrt{1-{r}^{2}}\mathrm{d}r\mathrm{d}\theta .$
and I leave the single variable integral to you.

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