곡선과 직선 사이 넓이

$\displaystyle f \left( x \right) = \frac{1}{4} x ^{3} + \frac{1}{2} x$, $g \left( x \right) = mx + 2$라 하고 두 함수 $y = f \left( x \right)$, $y = g \left( x \right)$의 그래프가 만나는 점의 $x$좌표를 $t$ $\left( 0 < t < 2 \right)$라 하면 주어진 그림에서 $\displaystyle A = \int _{0} ^{t} {} \left\{ g \left( x \right) - f \left( x \right) \right\} dx$, $\displaystyle B = \int _{t} ^{2} {} \left\{ f \left( x \right) - g \left( x \right) \right\} dx$ $\therefore$ $B - A$$\displaystyle = \int _{t} ^{2} {} \left\{ f \left( x \right) - g \left( x \right) \right\} dx - \int _{0} ^{t} {} \left\{ g \left( x \right) - f \left( x \right) \right\} dx$ $\displaystyle = \int _{t} ^{2} \left\{ f \left( x \right) - g \left( x \right) \right\} dx + \int _{0} ^{t} \left\{ f \left( x \right) - g \left( x \right) \right\} dx$ $\displaystyle = \int _{0} ^{2} \left\{ f \left( x \right) - g \left( x \right) \right\} dx$ $\displaystyle = \int _{0} ^{2} \left( \frac{1}{4} x ^{3} + \frac{1}{2} x - mx - 2 \right) dx$ $\displaystyle = \left[ \frac{1}{16} x ^{4} + \frac{1}{4} x ^{2} - \frac{m}{2} x ^{2} - 2 x \right] _{0} ^{2}$ $= 1 + 1 - 2 m - 4$ $= - 2 m - 2$ 따라서 $\displaystyle B - A = \frac{2}{3}$에서 $\displaystyle - 2 m - 2 = \frac{2}{3}$, $\displaystyle m = - \frac{4}{3}$