I am trying to understand N-spherical cap area formula (surface area of blue part), but it seems to give wrong answers.(adsbygoogle = window.adsbygoogle || []).push({});

for 1 dimensional cap obviously ## \frac{l_{cap}}{l_{sphere}}=\frac{l_{arc}}{l_{circle}}=\frac{r*θ}{r*2π}=\frac{θ}{2π} ##

But according to wikipedia formula ##\frac{l_{cap}}{l_{sphere}}=\frac{I(\frac{(2r-h)h}{r^2},\frac{n}{2},\frac{1}{2})}{2}##

https://en.wikipedia.org/wiki/Spherical_cap#Hyperspherical_cap

since ##h=r(1-cos(θ))##

## I(a,b,c)=\int_0^a (dx*x^{b-1}*(1-x)^{c-1})## (incomplete beta function)

and ## n=1##

##\frac{l_{cap}}{l_{sphere}}=\int_0^{sin^2(θ)} (\frac{dx}{\sqrt{x-x^2}})/2##

But

##\int_0^{sin^2(θ)} (\frac{dx}{\sqrt{x-x^2}})≠\frac{θ}{2π}##

Where is mistake?

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# Spherical cap

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