Triple integrals-finding the volume of cylinder.Find the volume of cylinder with base as the disk of unit radius in the xy plane centered at (1, 1, 0) and the top being the surface z = ( ( x − 1 ) 2 + ( y − 1 ) 2 ) 3 / 2 ..I just knew that this problem uses triple integral concept but dont know how to start. I just need someone to suggest an idea to start. I will proceed then.

Question

Triple integrals-finding the volume of cylinder.Find the volume of cylinder with base as the disk of unit radius in the xy plane centered at (1, 1, 0) and the top being the surface   z  =  (  (  x  −  1      )    2    +  (  y  −  1      )    2        )          3              /            2        ..I just knew that this problem uses triple integral concept but dont know how to start. I just need someone to suggest an idea to start. I will proceed then.

ri1men4dp · Accepted Answer

Step 1The volume of the cylinder C is given by the following triple integral  V  =      ∭          C        d  z  d  y  d  x  =      ∬          (      x      −      1              )        2            +      (      y      −      1              )        2            ≤      1        (  (  x  −  1      )    2    +  (  y  −  1      )    2        )          3              /            2        d  x  d  y    =      ∬                  X        2            +              Y        2            ≤      1        (      X    2    +      Y    2        )          3              /            2        d  X  d  Y  .Step 2Now use the polar coordinates   X  =  ρ  cos  ⁡  (  θ  ),   Y  =  ρ  sin  ⁡  (  θ  ). Then       X    2    +      Y    2    =      ρ    2   and   d  X  d  Y  =  ρ  d  ρ  d  θ where   ρ is the jacobian of the transformation. Thus we obtain  V  =      ∫          θ      =      0              2      π            ∫          ρ      =      0        1        ρ          3        (  ρ  d  ρ  d  θ  )  =  2  π  [      ρ    5        /    5      ]    0    1    =            2      π        5    .

Find the volume of cylinder with base as the disk of unit radius in the xy plane centered at (1,1,0) and the top being the surface z=((x-1)^2+(y-1)^2)^(3//2).

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