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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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.

Zane Barry · Accepted Answer

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    yAnd 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.

I need to find the volume of the intersection set A ∩ B whereby A

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