volume of a sphere

WBahn

Joined Mar 31, 2012
29,979
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.
 

WBahn

Joined Mar 31, 2012
29,979
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.
 

t_n_k

Joined Mar 6, 2009
5,455
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?
 

WBahn

Joined Mar 31, 2012
29,979
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.
 

t_n_k

Joined Mar 6, 2009
5,455
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.
 

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