정적분의 성질

함수 $f \left( x \right) = 3 x ^{2} - 16 x - 20$에 대하여 $\displaystyle \int _{- 2} ^{a} {} f \left( x \right) dx = \int _{- 2} ^{0} {} f \left( x \right) dx$ 이 성립하므로 $\displaystyle \int _{- 2} ^{a} {} f \left( x \right) dx + \int _{0} ^{- 2} {} f \left( x \right) dx = 0$ $\therefore$ $\displaystyle \int _{0} ^{a} {} f \left( x \right) dx = 0$ $\displaystyle \int _{0} ^{a} {\left( 3 x ^{2} - 16 x - 20 \right)} dx$$\displaystyle = \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} x ^{3} - 8 x ^{2} - 20 x \end{array} \right] _{0} ^{a}$ $= a ^{3} - 8 a ^{2} - 20 a$ $a ^{3} - 8 a ^{2} - 20 a = 0$에서 $a \left( a + 2 \right) \left( a - 10 \right) = 0$ $\therefore$ $a = 10$ $\left( \because a > 0 \right)$