Find the mass of the solid paraboloid Dequals={(r, thetaθ,z): 0less than or equals≤zless than or equals≤8181minus−r2, 0less than or equals≤rless than or equals≤99} with density rhorho(r, thetaθ,z)equals=1plus+StartFraction z Over 81 EndFraction z 81. Set up the triple integral using cylindrical coordinates that should be used to find the mass of the solid paraboloid as efficiently as possible. Use increasing limits of integration.

M = 5742π  

Step-by-step explanation:

Given:-

- Find the mass of a solid with the density ( ρ ):

                             ρ ( r, θ , z ) = 1 + z / 81

- The solid is bounded by the planes:

                             0 ≤ z ≤ 81 - r^2

                             0 ≤ r ≤ 9

Find:-

Find the mass of the solid paraboloid

Solution:-

- The mass (M) of any solid body is given by the following triple integral formulation:

- We can write the above expression in cylindrical coordinates:

- Perform integration:

- The mass evaluated is M = 5742π                      

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step-by-step explanation:

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step-by-step explanation: