Compite the integral ∫ V 4 − z 2 ( x 2 + y 2 ) 3 d x d y d z ,where V is the solid enclosed by the paraboloids x 2 + y 2 = z , x 2 + y 2 = 2 z and cones x 2 + y 2 = ( z − 2 ) 2 , x 2 + y 2 = 4 ( z − 2 ) 2 .

Question

Compite the integral      ∫    V              4      −              z        2                    (              x        2            +              y        2                    )        3                  d    x      d    y      d    z  ,where V is the solid enclosed by the paraboloids       x    2    +      y    2    =  z  ,      x    2    +      y    2    =  2  z and cones       x    2    +      y    2    =  (  z  −  2      )    2    ,      x    2    +      y    2    =  4  (  z  −  2      )    2    .

Antonio Dickerson · Accepted Answer

Let       D    −    =  {  (  r  ,  z  )  :  r  ≥  0  ,  z  ≤  2  ,  z  ≤      r    2    ≤  2  z  ,  (  z  −  2      )    2    ≤      r    2    ≤  4  (  z  −  2      )    2    }, and consider the following change of variables:   u  =  z      /        r    2   and   v  =  (  z  −  2      )    2        /        r    2  , then      |    det          (                                                                  ∂                u                                            ∂                r                                                                                        ∂                u                                            ∂                z                                                                                                        ∂                v                                            ∂                r                                                                                        ∂                v                                            ∂                z                                                        )        |    =  2            4      −              z        2                    r      5        .Then      ∭          V      ∩      {      z      ≤      2      }                  4      −              z        2                    (              x        2            +              y        2                    )        3              d  x  d  y  d  z  =  2  π      ∬                  D        −                        4      −              z        2                    r      6        r  d  r  d  z  =  2  π      ∫          u      =      1              /            2        1        ∫          v      =      1              /            4        1        1    2    d  u  d  v  =            3      π        8    .

Compite the integral ∫ V </msub> 4 −<!--

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