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 B whereby A = { ( x , y , z ) R | x 2 + y 2 + z 2 1 } and B = { ( x , y , z ) R | x 2 + y 2 1 / 2 } .
It is clear that A represents the unit ball centered at the origin and B represents the cylinder with radius 1 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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Answers (1)

Zane Barry
Answered 2022-07-08 Author has 5 answers
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 x 2 + y 2 1 / 2 1 x 2 y 2 d A = 2 x 2 + y 2 1 / 2 1 x 2 y 2 d x d y
And in polar coordinates
4 π 0 1 / 2 r 1 r 2 d r d θ .
and I leave the single variable integral to you.

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