이차함수 적분 공식

$f ( x ) = px ^{2} + qx + r ( \text{단} , p , q , r \text{은} \text{임의의} \text{실수} )$라 두면 $f ( 0 ) = r , f ( 1 ) = p + q + r , f ( 2 ) = 4 p + 2 q + r$이고 $\displaystyle \begin{aligned} \int _{0} ^{2} {} f ( x ) dx & = \int _{0} ^{2} {} ( px ^{2} + qx + r ) dx \end{aligned}$ $\displaystyle = \left[ \frac{p}{3} x ^{3} + \frac{q}{2} x ^{2} + rx \right] _{0} ^{2} = \frac{8 p}{3} + 2 q + 2 r$ $\displaystyle \begin{aligned} \frac{8}{3} p + 2 q + 2 r & = a r + b ( p + q + r ) + c ( 4 p + 2 q + r ) \end{aligned}$$= ( b + 4 c ) p + ( b + 2 c ) q + ( a + b + c ) r$ $\displaystyle b + 4 c = \frac{8}{3} , b + 2 c = 2 , a + b + c = 2$을 연립하면 $\displaystyle a = \frac{1}{3} , b = \frac{4}{3} , c = \frac{1}{3}$이므로 $\displaystyle \therefore a b c = \frac{8}{27}$