Volume of Intersection of cylinders (different radii)
I have been able to obtain the formula to calculate the volume like the solution to the link above.
My formula looked like this:
$$V = \int_{-r}^{r} \sqrt{r^2 -y^2}\sqrt{R^2 - y^2}dy$$
Where $r \leq R$
if $b = \frac{R}{r}$, how can I show $V = r^3 F(b)$.
I am having trouble finding an expression where volume is equal to $r^3$ times some expression that is only dependant on b
If you let $u=y/r$, you get the integral $$ V=\int_{-1}^1\sqrt{r^2-r^2u^2}\sqrt{R^2-r^2y^2}\,r\,du =r^3\int_{-1}^1\sqrt{1-u^2}\sqrt{R^2/r^2-u^2}\,du. $$ Since $b=R/r$, we find that $V=r^3F(b)$, with $$ F(b)=\int_{-1}^1\sqrt{1-u^2}\sqrt{b^2-u^2}\,du. $$ Note that we do not need to calculate $F$ explicitly.