구분구적법 넓이

[출제의도] 정적분과 무한급수의 관계 이해하기 $\displaystyle S _{k} = 1 - \frac{1}{2} \times \frac{k}{n} - \frac{1}{2} \times \frac{k}{n} \left( 1 - \frac{k}{n} \right)$ $\displaystyle = 1 - \frac{k}{n} + \frac{1}{2} \left( \frac{k}{n} \right) ^{2}$이므로 $\displaystyle \lim\limits _{n \rightarrow \infty} {\frac{1}{n} \sum\limits _{k = 1} ^{n - 1} S _{k}} = \lim\limits _{n \rightarrow \infty} {} \frac{1}{n} \sum\limits _{k = 1} ^{n - 1} \left\{ 1 - \frac{k}{n} + \frac{1}{2} \left( \frac{k}{n} \right) ^{2} \right\}$ $\displaystyle = \int _{0} ^{1} \left( 1 - x + \frac{1}{2} x ^{2} \right) dx$ $\displaystyle = \left[ x - \frac{1}{2} x ^{2} + \frac{1}{6} x ^{3} \right] _{0} ^{1}$$\displaystyle = \frac{2}{3}$ $\therefore$ $\displaystyle \alpha = \frac{2}{3}$ 따라서 $150 \alpha = 100$