Using a triple integral and cylindrical coordinates, calculate the volume of the solid bounded by the elliptic?

In order to use cylindrical coordinates, I think you mean
z = 9 - (x^2 + y^2) = 9 - x^2 - y^2 for the paraboloid.

Now, using cylindrical coordinates, the region is described by
z = 0 to z = 9 - r^2, r = 0 to 3, t = 0 to 2pi.

The integral is then given by
V = integral(t = 0 to 2pi, r = 0 to 3, z = 0 to 9 - r^2) 1 (r dz dr dt)
   = integral(t = 0 to 2pi, r = 0 to 3) r (9 - r^2) dr dt
   = integral(t = 0 to 2pi, r = 0 to 3) (9r - r^3) dr dt
   = integral(t = 0 to 2pi) (9 * 3^2/2 - 3^4/4) dt
   = (2pi) * (81/4)
   = 81pi/2.

I hope that helps!