Finding the volume of a solid bounded by a sphere and a paraboloidI am working on a problem that requires me to find the volume of the solid bounded by the sphere x 2 + y 2 + z 2 = 2 and the paraboloid x 2 + y 2 = z. I know that to do this, I must use triple integration.Thus far, I know that we need to reevaluate the given equations and put them into spherical coordinates in order to achieve our bounds. When we've gotten our bounds, it's just simple triple integration to computer the volume.So, for our sphere, I said that x 2 + y 2 + z 2 = 2 is equivalent to ρ = s q r t ( 2 ).Then, for our sphere, I get a little messed up. I'm trying to find ϕ boundaries here, but I get the equation ρ 2 s i n 2 ϕ = ρ c o s ϕ. Then, the ρ can cancel out, making it ρ s i n 2 ϕ = c o s ϕ, but my issue is that the ρ doesn't cancel out completely so I can't get the bounds for ϕ.

Question

Finding the volume of a solid bounded by a sphere and a paraboloidI am working on a problem that requires me to find the volume of the solid bounded by the sphere       x    2    +      y    2    +      z    2    =  2 and the paraboloid       x    2    +      y    2    =  z. I know that to do this, I must use triple integration.Thus far, I know that we need to reevaluate the given equations and put them into spherical coordinates in order to achieve our bounds. When we&#039;ve gotten our bounds, it&#039;s just simple triple integration to computer the volume.So, for our sphere, I said that       x    2    +      y    2    +      z    2    =  2 is equivalent to   ρ  =  s  q  r  t  (  2  ).Then, for our sphere, I get a little messed up. I&#039;m trying to find   ϕ boundaries here, but I get the equation       ρ    2    s  i      n    2    ϕ  =  ρ  c  o  s  ϕ. Then, the   ρ can cancel out, making it   ρ  s  i      n    2    ϕ  =  c  o  s  ϕ, but my issue is that the   ρ doesn&#039;t cancel out completely so I can&#039;t get the bounds for   ϕ.

Aidyn Mccarthy · Accepted Answer

Step 1You want the volume above the paraboloid below the sphere, so first observe that z goes from the paraboloid to the sphere. Let&#039;s use cylindrical coordinates:  x  =  r  cos  ⁡  θ  y  =  r  sin  ⁡  θ  z  =  zStep 2Then   d  V  =  r  d  z  d  r  d  θ and we have so far  ∫  ∫      ∫                  r                  2                                              2          −                      r                          2                                            r  d  z  d  r  d  θTo find the limits on r and   θ project the curve of intersection of the given surfaces, onto the   x  −  y plane. It is immediate that the region is a circle of radius 1 so we see that r goes from 0 to 1 and   θ goes from 0 to   2  π. Thus, we have      ∫          0              2      π            ∫          0              1            ∫                  r                  2                                              2          −                      r                          2                                            r  d  z  d  r  d  θ  =      π    6        (    8          2        −    7    )

Bodonimhk · Accepted Answer

Explanation:Cavalieri&#039;s principle.You want to compute the volume of the region that is generated by the rotation of   A  =      {    (    x    ,    y    )    ∈                  R            2        :    x    ∈    [    0    ,    1    ]    ,              x      2        ≤    y    ≤          2      −              x        2              }   around the y-axis, hence:  V  =  π      ∫          0              1        y    d  y  +  π      ∫          1                      2              (  2  −      y    2    )    d  y  =            π      6              (      8              2            −      7      )        .

Find the volume of the solid bounded by the sphere x^2+y^2+z^2=2 and the paraboloid x^2+y^2=z.

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