How would you find the volume of the solid bounded by the xy-plane, cylinder x^2+y^2=4, paraboloid z=(x^2+y^2)

Question

possible solutions...

a. 8p
b. 18p
c. 32p
d. 50p
e. 72p
f. 98p
g. 128p

or is it none of these ?

Answer 1

Using cylindrical coordinates the cylinder is r = 2 and the paraboloid is z = r^2
dV = rdrdθdz
Integrating θ from 0 to 2π,
dV = 2πrdrdz
Integrating r from z^1/2 to 2,
dV = (1/2)(2π)(2^2 - z) = π(4 - z)
Integrating z from 0 to 4,
V = π(4*4 - (1/2)4^2)
V = π(16 - 8)
V = 8π