PROBLEM: I am working on calculating volumes of geometric solids. All shapes have been pretty basic until now. I am bewildered on how to attack the problem of calculating the volume of a slice of a right circle cone.
VISUALIZING: The cone sits on a circular base with the apex directly above. The axis of symmetry passes through the apex and the center of the circle oriented normal to the circle. Now, when you pass a cutting plane parallel with the cone's axis of symmetry at a distance of R/2 (R being the cone's radius) between the circle's center and the circle's perimeter, a segment is 'cleaved' off. I am looking to determine the volume of that segment.
Welcome to the Math StackExchange. I've developed a framework for determining the volume of a hyperbolic slice off a cone. In a previous post (now deleted) I had an incorrect interpretation of the integration. Referring to the figure below, we see that radial slices through the cone form triangles in the volume to be determined. (Here we have assumed a cone of unit base radius without any loss in generality.) We can now develop the volume with a modified Pappus's ($2^{nd}$) Theorem. Briefly, the theorem states that the volume of a planar area of revolution is the product of the area, $A$ and path traced by it centroid, $C$. To wit, $V=2\pi CA$.
For bodied of non-circular or incomplete revolution we posit that
$$V=\int_{\theta_i}^{\theta_f} C(\theta)\ A(\theta)\ d\theta$$
Now, for a right triangle the centroid is located 1/3 of the base and 1/3 of the height from the right angle. Thus we have the area and centroid as follows:
$$ A(\theta)=\frac{b(\theta)\ h(\theta)}{2}\\ C(\theta)=(1-b(\theta))+b(\theta)/3=1-2b(\theta)/3 $$
There is still a lot to do here in order to determine $A$ and $C$, but I believe we established a sound approach to the problem.