how-to-prove-the-volumn-of-dimensional-sphare-formula-V-N-R-pi-N-2-N-2-1-R-N-

Question Number 97888 by M±th+et+s last updated on 10/Jun/20

h o w t o p r o v e ((t h e v o l u m n o f

d i m e n s i o n a l s p h a r e)) f o r m u l a

V_{N} (R) = \frac{π^{N / 2}}{Γ (\frac{N}{2} + 1)} R^{N}

Commented by EmericGent last updated on 10/Jun/20

I think you can integrate N times, I'm sorry, I can't try it now

Commented by M±th+et+s last updated on 10/Jun/20

t h a n k s f o r g i v i n g t h e i d e a m y b e i w i l l

t r y l a t e r

Answered by smridha last updated on 10/Jun/20

a t f i r s t c a l c u l a t e v_{N} (1) [v o l u m e o f u n i t c i r c l e i n N d i m]

t h e n m u l t i p y b y R^{N} .

y o u c a n u s e m u l t i d i m e n t i o n a l

c a l c u l u s a n d c o n s t r a c t e d J a c o b i a n

f o r N d i m e n t i o n .

y o u c a n s e e t h i s \dots

v o l u m e o f s p h e r e i n 3 D i s

\frac{4}{3} π R^{3} i n 2 D i t b e c a m e π R^{2} j u s t t h e

a r e a o f c i r c l e i n 1 D i t b e c a m e

2 R \dots \dots

{i t}^{'} s v e r y l e n g t h y p r o o f \dots

I c a n s o l v e t h i s \dots . t h a n k y o u \dots

Commented by M±th+et+s last updated on 10/Jun/20

t h a n k y o u v e r y m u c h s i r g o o d e x p l a i n i n g