Finding Volume Using Triple IntegrationSet up and evaluate a triple integral to find the volume of the region bounded by the paraboloid z = 1 − x 2 9 − y 2 100 and the xy-plane.I understand I'm finding the volume of a paraboloid that forms a "dome" over the xy-plane. Moreover, I can see the paraboloid intersects with the xy-plane to form an ellipse given by x 2 9 − y 2 100 = 1.I have tried setting this up using rectangular coordinates but the integral started looking extremely messy. I then tried spherical coordinates but had trouble find the upper bound of ρ. Specifically, I can't seem to successfully translate the rectangular equation z = 1 − x 2 9 − y 2 100 to a spherical equation and isolate ρ.

Question

Finding Volume Using Triple IntegrationSet up and evaluate a triple integral to find the volume of the region bounded by the paraboloid   z  =  1  −            x      2        9    −            y      2        100   and the xy-plane.I understand I&#039;m finding the volume of a paraboloid that forms a &quot;dome&quot; over the xy-plane. Moreover, I can see the paraboloid intersects with the xy-plane to form an ellipse given by             x      2        9    −            y      2        100    =  1.I have tried setting this up using rectangular coordinates but the integral started looking extremely messy. I then tried spherical coordinates but had trouble find the upper bound of   ρ. Specifically, I can&#039;t seem to successfully translate the rectangular equation   z  =  1  −            x      2        9    −            y      2        100   to a spherical equation and isolate   ρ.

mooseEredkefet8 · Accepted Answer

Explanation:If you do the change of variables       {                            X          =                      x            3                                                Y          =                      y            10                                                Z          =          z          ,                         then you shall have to compute the volume under the paraboloid   Z  =  1  −      X    2    −      Y    2

Finding Volume Using Triple Integration

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