Finding Volume of Rugby BallI am asked to find the volume of rugby ball whose surface is given by the ellipsoid: x 2 4 + y 2 4 + z 2 9 = 1I am having trouble figuring out which coordinate system I should use. Is it possible to solve the triple integral of the volume by just using cartesian co-ordinates, without making conversions to the spherical or cylindrical coordinate system?

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Finding Volume of Rugby BallI am asked to find the volume of rugby ball whose surface is given by the ellipsoid:            x      2        4    +            y      2        4    +            z      2        9    =  1I am having trouble figuring out which coordinate system I should use. Is it possible to solve the triple integral of the volume by just using cartesian co-ordinates, without making conversions to the spherical or cylindrical coordinate system?

kregde84 · Accepted Answer

Explanation:You can also find this volume by rotating the region inside the ellipse             y      2        4    +            z      2        9    =  1 about the z-axis:      V    =    2          ∫      0      3        π    (    R    (    z    )          )      2        d    z    =    2    π          ∫      0      3        4                  (              1    −                  z        2            9                      )              d    z    =    8    π                  [              z    −                  z        3            27                                ]                    0      3        =    16    π    .

Addyson Bright · Accepted Answer

Explanation:The shape is a unit sphere that has been scaled by factors of 2,2, and 3 in the x,y, and z directions. The volume is scaled by the same factors. So:  V  =  2  ⋅  2  ⋅  3  ⋅      4    3    π      1    3    =  16  π

Find the volume of rugby ball whose surface is given by the ellipsoid: x^2/4+y^2/4+z^2/9=1

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