Evaluate the iterated integral by converting to polar coordinates. \[ \int_{0}^{4} \int_{0}^{\sqrt{16-y^{2}}} 3 y d x d y \]

Step 1 ：The given integral is in Cartesian coordinates. To convert it to polar coordinates, we need to use the transformation equations x = rcos(θ) and y = rsin(θ).

Step 2 ：The limits of the integral also need to be changed accordingly. The given limits suggest that the region of integration is a semi-circle with radius 4 in the positive y half of the xy-plane. In polar coordinates, this corresponds to r ranging from 0 to 4 and θ ranging from 0 to π.

Step 3 ：The differential dx dy also changes to r dr dθ in polar coordinates. The integral becomes: \[\int_{0}^{\pi} \int_{0}^{4} 3 r^2 sin(θ) dr dθ\]

Step 4 ：Evaluating the integral gives a result of 384.

Step 5 ：Final Answer: The value of the iterated integral in polar coordinates is \(\boxed{384}\).