HP42s first major program (Double Integral) Best way to approach?

[DM48]

Today was a great day. I woke up to so many replies from people who are waaaay stronger in Math and programming than myself. Thank you to all. I sincerely mean it.

My DM42 arrived today as well. This is great news too.

I want to give a little more background on the problem.

This is a 1/4 turn rotation and this is why the final answer is multiplied by four.

The function calculates the volume of the material removed from a hollow right cylinder when a hole is drilled into the side.

Inside diameter = a
Outside diameter = b
Cored hole = c

Wall thickness is the difference between b and a.

I have to study through these responses and will come back. Many thanks again to everyone. I am learning to program the 42s and am under the gun to move this program from my 48 to the 42s.

I am looking to make this the fastest calculation possible with at least five, preferably six decimals.