Find the mass of the region of density g(x,y,z) = 1/(4 - x^2 - y^2) bounded by the paraboloid z = 4 - x^2 - y^2 and the​ xy-plane.?

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  • 2 months ago

    mass = ∫ ∫ ∫ g(x,y,z) dv.                        x^2 + y^2 = 4         ; x = cosø, y = sinø

            2pi   2   4- r^2

                 ∫   ∫  ∫.    1/(4 - r^2) rdz drø

               0   0.  0

           2pi  2                    4 - r^2

               ∫  ∫  r/(4 -r^2) (z)           drdø

              0  0                    0

              2pi.            2

         [ 0 ]      ( r^2/2).    =4pi

               0               0

    Myz = 0 Mz = 0

    Mxy = ∫ ∫ ∫ z g(x,y,z) dv.

            2pi.  2.  4- r^2

      =         ∫  ∫        ∫. 1/(4 - r^2) rdz drø

               0.  0.      0

               2pi       2                             4-r^2

      =            ∫ dø  ∫.   r/(4 - r^2) ( r^2/2).    dr

                 0.       0                             0

                    2pi.  2

    =          [ø ].     ∫. (1/2).r(4-r^2)dr

                    0.    0

                                         2

    =             pi(2r^2 -r^4/4)

                                         0

    =            pi ( 8 -4) = 4pi

    x= 0 . y= 0,  z = 4pi/4 = 1

    (x , y, z ) = ( 0, 0, 1)

  • rotchm
    Lv 7
    2 months ago

    M = ∫∫∫ ρ  dz dy dx, right?

    Answer me that, then plug in the expressions and show the result here.

    Then we will go into solving it. 

    Hopefully no one will spoil you the answer. That would be very irresponsible of them. 

    A lot of irresponsible people out there. 

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