Summing Deltas

Calculus Level 5

\[ \large \displaystyle \underset { n\rightarrow \infty }{ \lim } \frac { 1 }{ { n }^{ 2 } } \sum _{ k=1 }^{ n }{ { S }_{ k } } \]

Let \({ S }_{ k }\) denote area of triangle \({ AOB }_{ k }\) with 2 given sides of \( OA=1\), \({OB }_{ k }= k\) and \(\angle {AOB }_{ k }=\dfrac { k\pi }{ 2n }\) for positive integer \(k\).

If the value of the limit above can be expressed as \( C\times \pi^D \), where \(C\) and \(D\) are integers, find the value of \(C\times D\).

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