삼차곡선과 직선 넓이 조건

[출제의도] 정적분을 활용하여 수학 내적 문제 해결하기 $f ( x ) - g ( x ) = x ^{3} - 6 x ^{2} + 9 x - mx$$= x ( x - a ) ( x - b )$ $\displaystyle \int _{0} ^{b} {x ( x - a ) ( x - b ) dx}$ $\displaystyle = \left[ \frac{1}{4} x ^{4} - \frac{1}{3} ( a + b ) x ^{3} + \frac{1}{2} abx ^{2} \right] _{0} ^{b}$ $\displaystyle = \frac{1}{4} b ^{4} - \frac{1}{3} ( a + b ) b ^{3} + \frac{1}{2} ab ^{3}$ $\displaystyle = \frac{b ^{3}}{12} ( 2 a - b ) = 0$이므로 $b = 2 a$ $x \left\{ x ^{2} - 6 x + ( 9 - m ) \right\} = x ( x - a ) ( x - 2 a )$ $6 = 3 a$, $9 - m = 2 a ^{2}$이므로 $a = 2$, $m = 1$, $b = 4$ 따라서$\displaystyle \int _{a} ^{b} {\{ g ( x ) \} ^{2}} dx = \int _{2} ^{4} {x ^{2}} dx = \frac{56}{3}$