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Question Number 31141 by Cheyboy last updated on 03/Mar/18

$$\mathit{u}\mathit{s}\mathit{i}\mathit{n}\mathit{g}\mathit{t}\mathit{h}\mathit{e}\mathit{l}\mathit{i}\mathit{m}\mathit{i}\mathit{t}\mathit{d}\mathit{e}\mathit{f}\mathit{i}\mathit{n}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}$$$$\mathit{f}\mathit{i}\mathit{n}\mathit{d}\mathit{t}\mathit{h}\mathit{e}\mathit{a}\mathit{r}\mathit{e}\mathit{a}\mathit{o}\mathit{f}$$$$\mathit{f}\left(\mathit{x}\right)=\mathit{c}\mathit{o}\mathit{s}\left(\mathit{x}\right)[0,\pi /2]$$

Answered by Joel578 last updated on 03/Mar/18

$$A={\int }_0^{\frac{\pi }{2}}\cos xdx=\underset{n\to \mathrm{\infty }}{\lim }\underset{i=1}{\sum ^n}f\left(x_i\right)\mathrm{\Delta }{x}_i$$$$=\underset{n\to \mathrm{\infty }}{\lim }\underset{i=1}{\sum ^n}\cos \left(\frac{i\pi }{2n}\right)\frac{\pi }{2n}$$$$=\underset{n\to \mathrm{\infty }}{\lim }\frac{\pi }{2n}\underset{i=1}{\sum ^n}\cos \left(\frac{i\pi }{2n}\right)$$$$=\frac{\pi }{2}\underset{n\to \mathrm{\infty }}{\lim }\frac{\underset{i=1}{\sum ^n}\cos \left(\frac{i\pi }{2n}\right)}{n}$$$$=\frac{\pi }{2}\underset{n\to \mathrm{\infty }}{\lim }\frac{\cos \left(\frac{\pi }{2n}\right)+\cos \left(\frac{\pi }{n}\right)+\cos \left(\frac{3\pi }{2n}\right)+...+\cos \left(\frac{n\pi }{2n}\right)}{n}$$

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