Calculate the Volume of the unitary n-ball

Question

To do this first I need to prove that :

$$\displaystyle \int_R \sin^{n-2}\phi_1 \sin^{n-3}\phi_2\cdots\sin \phi_{n-2} d\theta d\phi_1 \cdots d\phi_{n-2} = \frac{2\pi^{n/2}}{\Gamma(n/2)}$$

where $ R=[0,2\pi] \times [0,\pi]^{n-2}$ but here there is a hint: Calculate the integral $\int_{\mathbb R^n}e^{-|x|^2}dx$ in spherical coordinates but this integral is a very complicated one, and from there I have found the following http://en.wikipedia.org/wiki/Volume_of_an_n-ball#Direct_integration_in_spherical_coordinates but How to do this without the betha function? so what Can be done with these two problems? Thanks a lot in advance

$$\int_{\mathbb{R}^n} e^{-|x|^2} dx = \left( \int_{-\infty}^{\infty} e^{-x^2} dx\right)^n$$ and I'm sure you know how to calculate the RHS for the special case $n = 2$. — achille hui, Apr 15 '15 at 15:32
You can also do this with induction on the dimension of the sphere (basis cases $n=1$ and $n=2$). — Chappers, Apr 15 '15 at 15:44
An how do that help in the calculation?, I want to calculate that integral in spherical coordinates, so I will have the jacobian and all that stuff, I have allready prove that the above integral is $\pi ^{n/2}$ but in spherical coordinates it is compkicated — user162343, Apr 15 '15 at 15:47
See http://math.stackexchange.com/questions/711082/volume-of-n-dimensional-ball and http://www.math.uwaterloo.ca/~karigian/teaching/multivariable-calculus/hypersphere.pdf for alternative ideas. — Martín-Blas Pérez Pinilla, Apr 15 '15 at 15:54
"How to do this without the betha function?" Try to find a recurrence formula for the integrals. — Martín-Blas Pérez Pinilla, Apr 24 '15 at 07:24

score 1 · Answer 1 · edited Apr 13 '17 at 12:20

1

Use the recurrence formula

$$ \int_{0}^{\pi/2}\sin^n(x)dx = \frac{n-1}n\int_{0}^{\pi/2}\sin^{n-2}(x)dx. $$

edited Apr 13 '17 at 12:20

Community

1

answered Apr 24 '15 at 07:33

Martín-Blas Pérez Pinilla

41,546
4
46
89

Calculate the Volume of the unitary n-ball

1 Answers1