The integral of the area of a circle ($\pi{r^2}$) is $\frac{1}{3}\pi{r^3}$, which is the volume of a cone when its height is equal to the radius of its base. It is my understanding that, in-keeping with the whole idea of calculus, you can view this relationship as infinitesimal circle areas, each with an infinitesimally smaller radius than the last, being stacked up on top of each other, forming a conic shape.
The original area was two dimensional, and through integration, it has been projected into three dimensions.
So, my question is, does the integral of the volume of a cone, which is $\frac{1}{12}\pi{r^4}$, describe the volume (or equivalent) of a shape composed of infinitesimally shrinking/enlarging cones being stacked on top of one another, projected in the fourth dimension?
Well, according to this Wiki page:
$$\text{V}_\text{n}\left(\text{R}\right)=\frac{\pi^\frac{\text{n}}{2}}{\Gamma\left(1+\frac{\text{n}}{2}\right)}\cdot\text{R}^\text{n}\tag1$$
So, when $\text{n}=4$:
$$\text{V}_4\left(\text{R}\right)=\frac{\pi^\frac{4}{2}}{\Gamma\left(1+\frac{4}{2}\right)}\cdot\text{R}^4=\frac{\pi^2}{2}\cdot\text{R}^4\tag2$$