Evaluating the Volume of a Cupola-Shaped Set by IntegrationLet f ( x , y ) = 1 − x 2 4 − y 2 and Ω = { ( x , y ) ∈ R 2 : f ( x , y ) ≥ 0 } .Compute the volume of the set A = { ( x , y , z ) ∈ R 3 : ( x , y ) ∈ Ω , 0 ≤ z ≤ f ( x , y ) } .My idea is to slice the set along the z-axis, obtaining a set E z - in fact, an ellipse - and computing the volume as ∫ 0 1 ∫ E z d x d y d z.However, I am stuck finding a way to describe E z . What is the best strategy to do that?

Question

Evaluating the Volume of a Cupola-Shaped Set by IntegrationLet   f  (  x  ,  y  )  =  1  −                    x        2            4        −      y    2   and   Ω  =  {  (  x  ,  y  )  ∈            R        2    :  f  (  x  ,  y  )  ≥  0  }  .Compute the volume of the set   A  =  {  (  x  ,  y  ,  z  )  ∈            R        3    :  (  x  ,  y  )  ∈  Ω  ,  0  ≤  z  ≤  f  (  x  ,  y  )  }  .My idea is to slice the set along the z-axis, obtaining a set       E    z   - in fact, an ellipse - and computing the volume as       ∫    0    1        ∫                  E        z              d  x  d  y  d  z.However, I am stuck finding a way to describe       E    z  . What is the best strategy to do that?

ahem37 · Accepted Answer

Step 1Use this parameterization for the whole 3 dimensional space                    x                            =        2        u        cos        ⁡        v                            y                            =        u        sin        ⁡        v                            z                            =        w                0  ≤  u  &amp;lt;  ∞  ,    0  ≤  v  &amp;lt;  2  π  ,    −  ∞  &amp;lt;  w  &amp;lt;  ∞then the integral for volume will be  V  =      ∫          v      =      0              2      π            ∫          u      =      0              1            ∫          w      =      0              1      −              u        2                        ∂      (      x      ,      y      ,      z      )              ∂      (      u      ,      v      ,      w      )        d  w    d  u    d  vStep 2Where             ∂      (      x      ,      y      ,      z      )              ∂      (      u      ,      v      ,      w      )        =      |                                                      ∂              x                                      ∂              u                                                                          ∂              x                                      ∂              v                                                                          ∂              x                                      ∂              w                                                                                      ∂              y                                      ∂              u                                                                          ∂              y                                      ∂              v                                                                          ∂              y                                      ∂              w                                                                                      ∂              z                                      ∂              u                                                                          ∂              z                                      ∂              v                                                                          ∂              z                                      ∂              w                                            |    =      |                            2          cos          ⁡          v                          −          2          u          sin          ⁡          v                          0                                      sin          ⁡          v                          u          cos          ⁡          v                          0                                      0                          0                          1                      |    =  2  u is the jacobian.

Let f(x,y)=1-x^2/4-y^2 and Omega={(x,y) in mathbb{R}^2:f(x,y) geq 0}. Compute the volume of the set A={(x,y,z) in mathbb{R}^3:(x,y) in Omega, 0 leq z leq f(x,y)}

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