Let $f(x)=ax^2+bx+c$, where $a∈R^+$ and $b^2-4ac<0$. Area bounded by y = f(x), x-axis and the lines x = 0, x = 1, is equal to:

$\frac{1}{12}(5f(1)-f(-1)+8f(0))$

f(0) = c; f(1) = a + b + c; f(-1) = a - b + c

$a=\frac{f(1)}{2}+\frac{f(-1)}{2}-f(0);b=\frac{(1)}{2}-\frac{f(-1)}{2};c=f(0)$

$⇒\int\limits_0^1(ax^2+bx+c)dx[\frac{ax^3}{3}+\frac{bx^2}{2}+cx]_0^1=\frac{a}{3}+\frac{b}{2}+c=\frac{1}{2}[5f(1)-f(-1)+8f(0)]$