Use cylindrical coordinates to evaluate the triple integral ∫∫∫E √(x^2+y^2) dV, where E is the solid bounded by the circular paraboloid z=4−16(x^2+y^2) and the xy -plane.

What I have so far is the thing being integrated turns into r^2 dr, and the limits are my problem, theta goes from 0 to 2pi, but I am unsure how to get the limits for r and z.

Try to draw a figure, to have an idea of the shape of E.

If you choose to integrate according to horizontal planes, then z varies between 0 and 4.
Express r in terms of z :
r = (1/4) √( 4 - z )

The bounds of your integral will be :
θ from 0 to 2π,
z from 0 to 4,
r from 0 to (1/4)√( 4 - z ).

End result, if I do no mistake, is 2π/15.