If I'm not crazy, area under the graph of the function $f$ between $x=a$ and $x=b$ is given by formula: $\displaystyle \int_a^b|f(x)|\mbox{d}x$. But I tried to count it for every polynomial and I got the wrong result. Is it difficult? My approach was that:
Let $f$ be any polynomial and $F'=f$. By integration by parts: $\int_a^b |f(x)|\mbox{d}x=\int_a^b f(x)\cdot \mbox{sgn}(f(x))\mbox{d}x= \\=F(x)\cdot\mbox{sgn}(f(x))\Big|_a^b - \int_a^b F(x)\cdot \mbox{sgn}'(f(x)) \mbox{d}x= F(x)\cdot\mbox{sgn}(f(x))\Big|_a^b$ because $\mbox{sgn}'(x)=0$ for all $x\in \mathbb{R}$.
But let $f(x)=x^2-5x+4$ and $a=0, \ b=5$, then $\int_a^b|f(x)|\mbox{d}x=\frac{49}{6}$ by WolframAlpha and with my method $F(x)\cdot\mbox{sgn}(f(x))\Big|_a^b=-\frac{5}{6}$.
Where I made a mistake?