Question

Use cylindrical or spherical coordinates, whichever seems more appropriate. Evaluate ∫∫∫E z dV, where E lies above the paraboloid
z = x2 + y2
and below the plane z = 2y. Use either the Table of Integrals or a computer algebra system to evaluate the integral.

Answer 1

Answer:

$\frac{160\pi }{3}$

Step-by-step explanation:

The region E that lies above the paraboloid : z = x^2 + y^2

while the region E that lies below the paraboloid : z = 4y

equate the two equations to get the representation of the curves

x^2 + y^2 = 4y = x^2 + ( y -2 )^2 = 4

the polar form will be : r = 4sin∅

attached below is the remaining part of the solution