The volume of the region between two spheres and the upper nappe of a coneI am really having trouble constraining the region between these three surfaces. I am imagining a sort of "Dome", or a "muffin head" sort of shape. Is this correct ? Anyway, I need to be able to write the following volume integral in rectangular, cylindrical and spherical coordinates:Consider the region that is between x 2 + y 2 + z 2 = 1, x 2 + y 2 + z 2 = 9, and finally above the upper nappe of the cone z 2 = 3 ( x 2 + y 2 ).Upon further consideration, does the smaller sphere even matter?

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The volume of the region between two spheres and the upper nappe of a coneI am really having trouble constraining the region between these three surfaces. I am imagining a sort of &quot;Dome&quot;, or a &quot;muffin head&quot; sort of shape. Is this correct ? Anyway, I need to be able to write the following volume integral in rectangular, cylindrical and spherical coordinates:Consider the region that is between       x    2    +      y    2    +      z    2    =  1,       x    2    +      y    2    +      z    2    =  9, and finally above the upper nappe of the cone       z    2    =  3  (      x    2    +      y    2    ).Upon further consideration, does the smaller sphere even matter?

Jackson Trevino · Accepted Answer

Step 1First find where the cone intersects the inner sphere:      x    2    +      y    2    =  1  −      z    2        z    2    =  3  −  3      z    2        z    2    =      3    4  Step 2This means that the radius of the boundary between the cone and inner sphere is       1    2  , and the radius of the boundary between the cone and the radius of the boundary for the outer sphere can be found to be             33        2  , using the same process. Converting the sphere equations into cylindrical coordinates and using as bounds   0  ≤  θ  &amp;lt;  2  π and       1    2    ≤  r  &amp;lt;            33        2  , the volume integral is as follows:  V  =      ∫    0          2      π            ∫                  1        2                                      33                2              (      9    −          r      2        −      1    −          r      2        )  r  d  r  d  θA similar procedure can be followed to find the integral in rectangular and spherical coordinates.

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