So, I came to this problem: to model a HALF cone volume using spherical coordinates in a way I can slice this half cone volume in any way I want and get a result.
After 3 weeks of no social life, I finally found the equation that rules the behavior of the slice, or, in other words, "Rho" ($\rho$). After finding $\rho$, the next step was to double integrate $(1/3)\rho^{3}\sin(\varphi)\mathrm{d}\varphi\mathrm{d}\theta)$ to get the volume I wanted.
The final equation was so tricky and I was so exhausted of the task that I had to use $1/3$ Composite Simpson Rule for Double Integrals and make 3,600 iterations to get close to the results, as I couldn't calculate the error (4th derivative was even harder to get).
Well, the job is done, client is satisfied, but the double integration is getting into my head and I don't believe I will ever see a swift and elegant equation of this half cone volume. But, who knows.
For the ones who want to dare to solve the equation and get the prize, here it is:
$$\frac{1}{3}\int_{0}^{\alpha}\int_{0}^{\frac{\pi}{2}}{\left(\frac{a\sin{\phi}+\sqrt{{\sin{\phi}}^2\left(a^2-bC\right)+C}}{1-b{\sin{\phi}}^2}\right)^3\sin{\phi}\mathrm{d}\phi\mathrm{d}\theta}$$
where as
$a = B\sin{\theta}-n\cos{\theta}$
$b = A{\sin{\theta}}^2$
*Attending Paul's request where $A=1+1/k^2,B=(nk-w)/k^2,C=w/k(w/k-n^2)$
** $R$ value in the circle equation is variable in Y-Axis
$A\ =\ 1+\frac{1}{k^2}$
$B\ =\ \frac{nk-w}{k^2}$
$C\ =\ \frac{w}{k}\left(\frac{w}{k}-2n\right)$
