큰 값 함수의 넓이

[출제의도] 정적분을 활용하여 문제해결하기 ⅰ) $f \left( x \right) \geq g \left( x \right)$일 때 $x ^{2} - 6 x + 10 \geq x$ $x \leq 2$ 또는 $x \geq 5$ $\left| f \left( x \right) - g \left( x \right) \right| = f \left( x \right) - g \left( x \right)$이므로 $\displaystyle h \left( x \right) = \frac{\left| f \left( x \right) - g \left( x \right) \right| + f \left( x \right) + g \left( x \right)}{2} = f \left( x \right)$ ⅱ) $f \left( x \right) < g \left( x \right)$일 때 $x ^{2} - 6 x + 10 < x$ $2 < x < 5$ $\left| f \left( x \right) - g \left( x \right) \right| = - f \left( x \right) + g \left( x \right)$이므로 $\displaystyle h \left( x \right) = \frac{\left| f \left( x \right) - g \left( x \right) \right| + f \left( x \right) + g \left( x \right)}{2} = g \left( x \right)$ $\displaystyle \therefore h \left( x \right) = {\begin{cases} x ^{2} - 6 x + 10 _{{} _{{} _{}}} & \left( x \leq 2 \text{또는} x \geq 5 \right) _{{} _{{} _{}}} \\ x ^{{} ^{{} ^{}}} & \left( 2 < x < 5 \right) ^{{} ^{{} ^{}}} \end{cases}}$ 구하는 넓이는 $\displaystyle \int _{0} ^{2} {\left( x ^{2} - 6 x + 10 \right) dx} + \int _{2} ^{4} {xdx}$ $\displaystyle = \left[ \frac{1}{3} x ^{3} - 3 x ^{2} + 10 x \right] _{0} ^{2} + \left[ \frac{1}{2} x ^{2} \right] _{2} ^{4} = \frac{50}{3}$