급수와 정적분

[출제의도] 정적분과 급수의 합 사이의 관계를 이해한다. $\displaystyle \lim\limits _{n \rightarrow \infty} {\sum\limits _{k = 1} ^{n} \frac{k \pi}{n ^{2}} f \left( \frac{\pi}{2} + \frac{k \pi}{n} \right)}$$\displaystyle = \frac{1}{\pi} \int _{0} ^{\pi} {xf \left( \frac{\pi}{2} + x \right) dx}$ $\displaystyle = \frac{1}{\pi} \int _{0} ^{\pi} {x \cos \left( \frac{\pi}{2} + x \right) dx}$$\displaystyle = \frac{1}{\pi} \int _{0} ^{\pi} {x \left( - \sin x \right) dx}$ $\displaystyle = \frac{1}{\pi} \left( \begin{array}{l} \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} x \cos x \end{array} \right] _{0} ^{\pi} - \int _{0} ^{\pi} {\cos xdx} \end{array} \right)$ $\displaystyle = \frac{1}{\pi} \left( \begin{array}{l} - \pi - \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} \mathrm{\sin} \mathit{x} \end{array} \right] _{0} ^{\pi} \end{array} \right)$$\displaystyle = \frac{1}{\pi} \times ( - \pi )$$= \mathit{-} 1$