I found the equation of the boundary circle by setting z to 4 in the paraboloid.

Then I did some work to get polar coords:

$\displaystyle x^2+y^2 = 1$

$\displaystyle x^2+y^2 = r^2$

$\displaystyle 1-x^2-y^2 = 1-r^2$

Then I set up my integral as such

$\displaystyle \int_0^{2\pi}\int_{0}^{1}(1-r^2)rdrd\theta$

edit: I used the wrong integrand. It should be 4r-4r^3.