합성함수 정적분

$\displaystyle f ( x ) = \begin{cases} 32 x ( 0 \leq x \leq 1 ) \\ \\ - 32 x + 64 ( 1 \leq x \leq 2 ) \end{cases}$ $\displaystyle g ( x ) = \begin{cases} - 2 x + 1 \left( 0 \leq x \leq \frac{1}{2} \right) \\ \\ \frac{2}{3} x - \frac{1}{3} \left( \frac{1}{2} \leq x \leq 2 \right) \end{cases}$ ⅰ) $\displaystyle 0 \leq x \leq \frac{1}{2}$일 때, $f ( g ( x ) ) = f ( - 2 x + 1 ) = 32 ( - 2 x + 1 ) = - 64 x + 32$ ⅱ) $\displaystyle \frac{1}{2} \leq x \leq 2$일 때, $\displaystyle f ( g ( x ) ) = f \left( \frac{2}{3} x - \frac{1}{3} \right) = 32 \left( \frac{2}{3} x - \frac{1}{3} \right) = \frac{64}{3} x - \frac{32}{3}$ $\displaystyle \begin{aligned} & \frac{1}{3} \int _{0} ^{2} {} f ( g ( x ) ) dx \end{aligned}$ $\displaystyle = \frac{1}{3} \left\{ \int _{0} ^{\frac{1}{2}} {} ( - 64 x + 32 ) dx + \int _{\frac{1}{2}} ^{2} {} \left( \frac{64}{3} x - \frac{32}{3} \right) dx \right\}$ $\displaystyle = \frac{1}{3} \left\{ \left[ - 32 x ^{2} + 32 x \right] _{0} ^{\frac{1}{2}} + \left[ \frac{32}{3} x ^{2} - \frac{32}{3} x \right] _{\frac{1}{2}} ^{2} \right\}$ $\displaystyle = \frac{1}{3} ( 8 + 24 ) = \frac{32}{3} = 10.666 \cdots$ $\therefore 10.67$