Finding The Volume Of a Shape That Is Given By the Formula 3 x 2 + 2 y 2 + z 2 ≤ 6How can I find the volume of the shape that is given by the formula 3 x 2 + 2 y 2 + z 2 ≤ 6

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Finding The Volume Of a Shape That Is Given By the Formula   3      x    2    +  2      y    2    +      z    2    ≤  6How can I find the volume of the shape that is given by the formula  3      x    2    +  2      y    2    +      z    2    ≤  6

Siena Bennett · Accepted Answer

Step 1Dividing both sides by 6, this is an ellipsoid in standard form with semi-axes of length       6  ,       3  , and       2  . It is well-known that the volume of an ellipsoid   E  :            x      2              a      2        +            c      2              b      2        +            z      2              c      2        =  1has volume       4    3    π  a  b  c, so in this case your answer is                               6                    .Step 2If you are interested in actually carrying out the integration, then consider rewriting the equation in the above form and then performing the following change of coordinates:      {                            x          =                      2                    r          cos          ⁡          θ          sin          ⁡          ϕ                                      y          =                      3                    r          sin          ⁡          θ          sin          ⁡          ϕ                                      z          =                      6                    cos          ⁡          ϕ                        The new bounds of integration will take r from 0 to 1,   θ from 0 to   2  π, and   ϕ from 0 to   π. The resulting Jacobian from the change of variables will be   a  b  c  =  6. The remaining steps are pretty straightforward, so I&#039;ll leave that to you.

Finding The Volume Of a Shape That Is Given By the Formula 3x^2+2y^2+z^2 leq 6.

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