Use cylindrical coordinates.
25. (a) Find the volume of the region E that lies between the paraboloid z = 24 - x2 - y2 and the cone Z = 2 .
(b) Find the centroid of E (the center of mass in the case where the density is constant).
To find: The volume of the given solid by using cylindrical coordinates.
The region E lies between the paraboloid and the cone .
If is a cylindrical region E given by where , then,
The cylindrical coordinates corresponding to the rectangular coordinates is,
From the given equations, the value of z is obtained below.
The above equation is a quadratic equation.
Thus, solve the equation and obtain the value of .
Therefore, r varies from 0 to 4, and varies from 0 to and z varies from 2r to .
Use the formula mentioned above to change the given problem into cylindrical coordinates. Then, by the equation (1), the volume of the given solid is computed as follows.
Integrate with respect to z and apply the limit of it
To find: The centroid of the given region E.
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