Reading https://en.wikipedia.org/wiki/Pi#Definition I stumpled upon the following definition as an integral, presumably given by Weierstrass:$$\pi = \int_{-1}^1 \frac{dx}{\sqrt{1-x^2}}$$However I don't understand, why one would use such a definition. Starting from the equation for a circle with radius one $x^2+y^2=1$ we get $y = \pm\sqrt{1-x^2}$ and a much more natural definition of $\pi$ would be two times the area under this curve like$$\pi = 2\int_{-1}^1 \sqrt{1-x^2}\, dx$$Why did Weierstrass instead choose to give an "inverted" definition?
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