Find the volume of a solid which is bounded by the paraboloid 4 z = x 2 + y 2 , the cone z 2 = x 2 + y 2 and the cylinder x 2 + y 2 = 2 xI approached this problem by trying to find the volume bounded by the paraboloid and the cylinder and then subtracting it from the volume bounded by the cone and the cylinder. But I am getting the wrong answer. I converted the all the bounds into cylindrical co-ordinates.For finding the volume bounded by the cone and the cylinder,Bounds of integration: z = 0 to z = r , r = 0 to r = 2 cos ⁡ ( θ ) , θ = 0 to θ = 2 π .For finding the volume bounded by the paraboloid and the cylinder,Bounds of integration: z = 0 to z = r 2 4 , r = 0 to r = 1 , θ = 0 to θ = 2 π .

Question

Find the volume of a solid which is bounded by the paraboloid   4  z  =      x    2    +      y    2  , the cone       z    2    =      x    2    +      y    2   and the cylinder       x    2    +      y    2    =  2  xI approached this problem by trying to find the volume bounded by the paraboloid and the cylinder and then subtracting it from the volume bounded by the cone and the cylinder. But I am getting the wrong answer. I converted the all the bounds into cylindrical co-ordinates.For finding the volume bounded by the cone and the cylinder,Bounds of integration:      z  =  0    to      z  =  r        ,        r  =  0    to      r  =  2  cos  ⁡  (  θ  )        ,        θ  =  0    to      θ  =  2  π   .For finding the volume bounded by the paraboloid and the cylinder,Bounds of integration:      z  =  0    to      z  =            r      2        4          ,        r  =  0    to      r  =  1        ,        θ  =  0    to      θ  =  2  π   .

kilinumad · Accepted Answer

Step 1Your bounds for   θ are wrong;   θ should run from   −      π    2   to       π    2  :  V  =      ∫          −              π        2                            π        2                  ∫          0              2      cos      ⁡      θ            ∫                            r          2                4              r    r    d  z    d  r    d  θ  =      32    9    −      3    8    π  .Step 2If you want to do it by subtracting the volume bouded by the paraboloid to the one bouded by the cone, then                     V                            =                  ∫                      −                          π              2                                                          π              2                                                ∫                      0                                2            cos            ⁡            θ                                    ∫                      0                    r                r                d        z                d        r                d        θ        −                  ∫                      −                          π              2                                                          π              2                                                ∫                      0                                2            cos            ⁡            θ                                    ∫                      0                                                              r                2                            4                                      r                d        z                d        r                d        θ        =                                          =                  32          9                −                  3          8                π        .

Find the volume of a solid which is bounded by the paraboloid 4z=x^2+y^2, the cone z^2=x^2+y^2 and the cylinder x^2+y^2=2x.

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