Here's what I get using both formulations and both values of R:

 

Oh, the two equations are consistent with each other. There is an inconsistency between them and the formulas I set up in my spreadsheet. I think the ones above are correct, but I'm not positive unless I go through it from scratch. When I get some time this evening I will resolve the problem and find out where I made my mistake. I'm pretty sure it will be in the spreadsheet.

 

Yep. Looked at my equations in the spreadsheet and sure enough I found the mistake right away. When I calculated the value of y where the sphere intersects the z=-2unit plane, my formula computed it using R, x, and z. But later I added a little r cell that matches the R used in the formulas I posted but I changed the formula y to just use that r instead of R without updating the formula to reflect the fact that x has already been taken into account.

Now life is good.

 

I thought I'd try to solve the problem using the Cartesian System ...

Unfortunately I can't produce the same answer given earlier.

I have the "solution" as the integration shown below ...

\(\text{V=\int^{+1}_{-1}\{ \frac{(R^2-x^2)}{2}\[\pi - arcsin\(\sqr{\frac{12-x^2}{R^2-x^2}}\)\]+\sqr{12-x^2} \} \ dx \ = 39.7385 \ cu. \ units}\)

Did anyone else try the Cartesian approach?

 

I thought I'd try to solve the problem using the Cartesian System ...

Unfortunately I can't produce the same answer given earlier.

I have the "solution" as the integration shown below ...

\(\text{V=\int^{+1}_{-1}\{ \frac{(R^2-x^2)}{2}\[\pi - arcsin\(\sqr{\frac{12-x^2}{R^2-x^2}}\)\]+\sqr{12-x^2} \} \ dx \ = 39.7385 \ cu. \ units}\)

Did anyone else try the Cartesian approach?

Check your units.

 

Check your units.

Thanks for the "heads up" on the units. Although it looks dubious, I believe the "discrepancy" lies in my having partially substituted some constant (linear) dimensions specific to the problem during the formulation of the result.

FYI I have attached my full derivation for critiquing by anyone who might be interested.

 

Did anyone else try the Cartesian approach?

I did it like this:

 

OK. Many thanks for the response.