Finding the volume of the intersection region of two spheresWe have two spheres whose radiuses are a cm each and their centre coordinates are (0, a, 0) and (0, 0, 0) consecutively.How can we determine the volume of the intersection regions of these given two spheres by using triple integrals?

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g5riem7z · Accepted Answer

Step 1First note that the two spheres have equations:      x    2    +      y    2    +      z    2    =      a    2          and          x    2    +  (  y  −  a      )    2    +      z    2    =      a    2   and intersect on the plane   y  =  a      /    2.So the volume of the intersection region is made of two identical cups that have height   h  =  a      /    2 and radius of the basis circle   b  =            3        2    a and we can use the formula for the volume of a cup to evaluate the volume as:  V  =  2      V          c      u      p        =  2  ⋅            π      a        12        (          9      4              a      2        +                  a        2            4        )    =  2  ⋅            5      π              a        3              24  If you want to justify this result with an integral, you can do it with a simple integral noting that the single cup is a solid of revolution around the y axis of the curve   z  =            a      2        −          y      2       in the   y  −  z plane. So the volume is:      V          c      u      p        =      ∫          a              /            2        a    π  z  (  y      )    2    d  y  =  π      ∫          a              /            2        a    (      a    2    −      y    2    )  d  y  =            5      π        24        a    3  If you want necessarily a triple integral than the better is to note the cylindrical symmetry around the y axis and use cylindrical coordinates around it ( this means that we switch the name of y and z axis), so the volume of a single cup becomes:       V          c      u      p        =      ∫          a              /            2        a        ∫    0          2      π            ∫    0                            a          2                −                  y          2                      ρ    d  ρ    d  φ    d  ythat, with a bit of work, gives the same result.The use of a triple integral in rectangular coordinates is possible but really masochistic.

We have two spheres whose radiuses are a cm each and their centre coordinates are (0,a,0) and (0,0,0) consecutively. How can we determine the volume of the intersection regions of these given two spheres by using triple integrals?

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