I tried to derivate pi using calculus but i just found a quite different result. Can you spot my wrong please?(adsbygoogle = window.adsbygoogle || []).push({});

First i started with equlation of a circle which is:

$$x^2+y^2=r^2$$

I am assuming circle's center stands on the center of origin.

To reach pi we shoud consider the situation that diameter = 1 because when diameter equals 1 circumference equals ##\pi##.Then:

$$x^2+y^2=\frac{1}{4}$$

Simplify the equlation to get:

$$y=\sqrt{\frac{1}{4} - x^2}$$

We can use the formula which says that our arc lenght is

$$ L =\int_a^b \sqrt{1-(f^\prime(x))^2}$$

Take the derivative of our function which is:

$$-\frac{x}{\sqrt{\frac{1}{4} - x^2}}$$

Stick this equlation to the formula to found arclenght.(and multiply by two to get full arclenght we just find the half of the circle)

$$L=\int_a^b \sqrt{1-\frac{4x^2}{1 -4x^2}}$$

The circumference is the twice of the value of this integral between ##-\frac{1}{2}## and ##\frac{1}{2}##.

So the equlation comes to:

$$L=\int_{-\frac{1}{2}}^\frac{1}{2} \sqrt{1-\frac{4x^2}{1 -4x^2}}$$

That's where i got stuck. So i used our friend Wolfram Alpha to reach result(taking indefinite integral):

$$\frac{1}{2} E(sin^{-1}(2x)|2)$$

I've absolutely no idea what that means and i simply take the definite integral and found the result as(by using the WolframAlpha of course):

$$E(2)$$

Well i have no idea what does this mean and i ask for some help from you. Can someone explain what this result is and how did i get to this result? By the way E(2) equals:

$$

0.59907011736779610371996124614016193911360633160782577913... +

0.59907011736779610371996124614016193911360633160782577913... i

$$

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# I Derivation of pi using calculus

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