Unit circle/Integral of square root of 1-x^2/Example

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The upper curve of the unit circle is the set

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For a given , , there exists exactly one fulfilling this condition, namely . Hence, the area of the upper half of the unit circle is the area beneath the graph of the function , above the interval , that is

Applying substitution with

(where is bijective, due to), we obtain, using example, the identities

In particular, we get that

is a primitive function for . Therefore,