Use cylindrical coordinates.
18. Evaluate where E is enclosed by the paraboloid z = x2 + y2 and the plane z = 4.
To evaluate: The given triple integral by using cylindrical coordinates.
The function is .
The region E is enclosed by the paraboloid and the plane .
If is a cylindrical region E given by where , then,
If is the function of and is the function of and is the function of z, then,
The cylindrical coordinates corresponds to the rectangular coordinates is,
Solve the given equations,
It is observed that r varies from 0 to 2, varies from 0 to and z varies from to 4. Use the formula mentioned above to change the given problem into cylindrical coordinates. Then, by the equation (1), the value of the given triple integral is,
Use the equation (2) to integrate and apply the limit values
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