Hi everyone! I want to calculate the formula of hypervolume (I label it "H") of 4D hypercone, which consists of 3D base which is ball with radius R, apex and lines from apex to points of ball that are not in the same 3D-space as ball. Let's label its height as "h". If 3D volume can be calculated through definite integral:

[tex]V=\int\limits_{a}^{b}S(x)dx[/tex]

I think that hypervolume can be calculated:

[tex]H=\int\limits_{a}^{b}V(x)dx[/tex]

Cross-section which is parallel to base is also ball with radius r(x), and distance between apex and such cross-section is x. Radiuses and height make a triangle:

As full and partial triangles are similar:

[tex]\frac{x}{h}=\frac{r(x)}{R} \Rightarrow r(x)=\frac{Rx}{h}[/tex]

So hypervolume is:

[tex]H=\int\limits_{0}^{h}{V(x)dx}=\int\limits_{0}^{h}\frac{4\pi r^3(x) dx}{3}=\frac{4\pi}{3}\int\limits_{0}^{h}{r^3(x)dx}=\frac{4\pi}{3}\int\limits_{0}^{h}{\frac{R^3 x^3}{h^3}}=\frac{4\pi R^3}{3 h^3}\int\limits_{0}^{h}{x^3 dx}=\frac{4\pi R^3}{3 h^3} \frac{x^4}{4}\begin{array}{|l} h \\ 0 \end{array}=\frac{4\pi R^3 h^4}{3 \cdot 4 h^3}=\frac{\pi R^3 h}{3}[/tex]

Is this formula correct?