Find Length of Sine Curve using Calculus

[1994Bhaskar]

I tried to find the length of a sine curve using calculus.I got stuck in the integral of integral(sqrt(cos(x)^2+1), x, 0, a). Limits are from 0 to point a,i.e. length of curve from 0 to any point a.With some approximations I found out the length of sine curve as a result.I have attached the derivation of my result.I tried searching books and internet for this result.But couldn't find like this one.It also seems that this result gives more precise answer.

 

[chiro]

Hey 1994Bhaskar.

I cheated and used the Wolfram Alpha computational knowledge engine and the answer according to them, was that the answer was in the form of an elliptic integral of the 2nd kind. On wikipedia, it gives you this:

http://en.wikipedia.org/wiki/Elliptic_integral#Incomplete_elliptic_integral_of_the_second_kind

If you want more information go to Wolfram alpha and type out 'int (1 + (cos(x))^2)^(1/2) dx' to get more information.

 

[1994Bhaskar]

Hey 1994Bhaskar.

I cheated and used the Wolfram Alpha computational knowledge engine and the answer according to them, was that the answer was in the form of an elliptic integral of the 2nd kind. On wikipedia, it gives you this...

I know elliptic integrals.It comes when you try to find time period of a simple pendulum without taking usual approximation:that angular displacement is very small and that
sin(θ)≈θ in radians.But even elliptical integrals can't be solved analytically.You have to solve it in a calculator/computer or use interpolation formula's to carry out integral approximately.However the method of approximation which I used gives a result which gives better answer compared to a calculator or interpolation.In the end of pdf I have given three checks of that result which I haven't seen anywhere.I have also searched the net extensively.That's why I posted here to ask you guys if it's a new result or not??

 

[chiro]

I know elliptic integrals.It comes when you try to find time period of a simple pendulum without taking usual approximation:that angular displacement is very small and that
sin(θ)≈θ in radians.But even elliptical integrals can't be solved analytically.You have to solve it in a calculator/computer or use interpolation formula's to carry out integral approximately.However the method of approximation which I used gives a result which gives better answer compared to a calculator or interpolation.In the end of pdf I have given three checks of that result which I haven't seen anywhere.I have also searched the net extensively.That's why I posted here to ask you guys if it's a new result or not??

I don't know if its a new result or not (hell I didn't know what an elliptic integral was!). Hopefully somebody else can give you a better answer.

 

[1994Bhaskar]

I don't know if its a new result or not (hell I didn't know what an elliptic integral was!). Hopefully somebody else can give you a better answer.

Thanks for your effort.It means a lot.Can you also suggest where else can I put up my query? Or mail someone?
Any help will be useful.