구간별 함수의 미분가능조건과 리만합의 극한

$g _{1} ( x ) = 4 x - 3$, $g _{2} ( x ) = ax ^{2} + bx$라 하면 $f ( x )$가 열린구간 $( 0 , 2 )$에서 미분가능하므로 $g _{1} ( 1 ) = g _{2} ( 1 )$이고 $g _{1}' ( 1 ) = g _{2}' ( 1 )$이다. 따라서 $a + b = 1$, $4 = 2 a + b$에서 $a = 3$, $b = - 2$ ∴ $g _{2} ( x ) = 3 x ^{2} - 2 x$ $\displaystyle \lim\limits _{n \rightarrow \infty} \frac{1}{n} \sum\limits _{k=1} ^{n} f \left( \frac{2 k}{n} \right)$$\displaystyle = \frac{1}{2} \lim\limits _{n \rightarrow \infty} \frac{2}{n} \sum\limits _{k=1} ^{n} f \left( \frac{2 k}{n} \right)$ $\displaystyle = \frac{1}{2} \int _{0} ^{2} f ( x ) dx$ $\displaystyle = \frac{1}{2} \left\{ \int _{0} ^{1} f ( x ) dx + \int _{1} ^{2} f ( x ) dx \right\}$ $\displaystyle = \frac{1}{2} \left\{ \int _{0} ^{1} ( 4 x - 3 ) dx + \int _{1} ^{2} ( 3 x ^{2} - 2 x ) dx \right\}$ $\displaystyle = \frac{1}{2} \left\{ \left[ 2 x ^{2} - 3 x \right] _{0} ^{1} + \left[ x ^{3} - x ^{2} \right] _{1} ^{2} \right\}$ $\displaystyle = \frac{3}{2}$