역함수 급수 정적분

$\displaystyle \lim\limits _{n \rightarrow \infty} {\sum\limits _{k = 1} ^{n} g \left( \frac{3 k}{n} \right) \frac{1}{n}} = \frac{1}{3} \int _{0} ^{3} {g ( x ) dx}$ 한편, $\displaystyle \int _{0} ^{3} {g ( x ) dx} = 9 - \int _{0} ^{3} {f ( x ) dx}$ $\displaystyle \left( \because \frac{x ^{3}}{9} = x , x = 3 \right)$ $\displaystyle \frac{1}{3} \int _{0} ^{3} {g ( x ) dx}$$\displaystyle = \frac{1}{3} \left\{ 9 - \int _{0} ^{3} {f ( x ) dx} \right\}$$\displaystyle = \frac{1}{3} \left\{ 9 - \left[ \frac{x ^{4}}{36} \right] _{0} ^{3} \right\} = \frac{9}{4}$