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

[DM48]

(05-27-2020 02:13 AM)Thomas Okken Wrote:  The HP-42S doesn't have expressions, so you'd have to write a keystroke program instead, like this:

Code:

00 { 40-Byte Prgm }
01▸LBL "FN"
02 MVAR "R"
03 MVAR "C"
04 MVAR "X"
05 RCL "R"
06 RCL "C"
07 X↑2
08 RCL "X"
09 X↑2
10 -
11 SQRT
12 ×
13 RCL "R"
14 X↑2
15 RCL "X"
16 X↑2
17 -
18 SQRT
19 ÷
20 END

HOWEVER... The HP-42S and DM42 don't support double integrals. If you try to use INTEG inside a function that is itself being integrated, you'll get the error message Integ(Integ).

I am trying to enter in this expression that Albert Chan supplied (tested this on my 48G and it works 33% faster than the double integral, thanks Albert!)

First Question

When entering the integral, how do you know what the calculator interprets to be under the square root house without using parentheses? I am use to the 48G Equation Writer to make sure I have crazy equations like this formatted correctly when entering in a program. The second part of Albert's expression has me puzzled how to key this in with certainty. Any documentation on this I can read or is it just a RPN Order of Operations thing?

\(I = \int _0 ^c \sqrt{c^2-x^2}\left( \sqrt{b^2-x^2} - \sqrt{a^2-x^2} \right)\;dx\)