- #1
yucheng
- 232
- 57
- Homework Statement
- Find the volume of a 4-space ball, with radius R.
(Alternatively you can integrate this via Cavalieri’s principle, by using the fact that the crosssectional volume is given by ##\frac{4}{3} \pi r^3##
, the volume of a 3-ball)
See example at end of page 1 of https://www.math.ucla.edu/~bonsoon/math32bh.winter2020/notes/short_notes_-_More_on_Change_of_variables_formula.pdf
- Relevant Equations
- N/A
I tried integrating the 4-volume of a 4-hemisphere, that is, $$\int^{R}_{0} \frac{4}{3} \pi r^3 dw$$ (along w-axis), since ##r## is proportional to ##w##, where ##r=\frac{w}{R} R##, ##r=w##, thus the integral becomes $$\int^{R}_{0} \frac{4}{3} \pi w^3 dw = \frac{\pi}{3} R^4$$ The volume of a 4-D hypersphere is ##\frac{2\pi}{3} R^4##
Clearly, something is not right.
Clearly, something is not right.