X軸と二次凾数で囲まれる面積

$y=0$(X軸)と$(x-\alpha )(x-\beta )=0$(二次凾数)とで囲まれる領域の面積

$x^2-(\alpha +\beta )x+\alpha \beta =0(\alpha <\beta )$であり$x^2$の係数が正で，X軸の下側に面積ができる場合(下に凸)
$$\begin{array}{rcl}{\displaystyle {\int }_{\alpha }^{\beta }(x-\alpha )(x-\beta )\mathrm{d}x}& =& {\displaystyle {\int }_{\alpha }^{\beta }(x-\alpha )(x-\beta -\alpha +\alpha )\mathrm{d}x}\\ & =& {\displaystyle {\int }_{\alpha }^{\beta }(x-\alpha )((x-\alpha )-(\beta -\alpha ))\mathrm{d}x}\\ & =& {\displaystyle {\int }_{\alpha }^{\beta }(x-\alpha {)}^2-(\beta -\alpha )(x-\alpha )\mathrm{d}x}\\ & =& {\displaystyle {\int }_{\alpha }^{\beta }(x-\alpha {)}^2\mathrm{d}x-{\int }_{\alpha }^{\beta }(\beta -\alpha )(x-\alpha )\mathrm{d}x}\\ & =& {\displaystyle {\left[\frac{(x-\alpha {)}^3}{3}\right]}_{\alpha }^{\beta }-{\left[\frac{(\beta -\alpha )(x-\alpha {)}^2}{2}\right]}_{\alpha }^{\beta }}\\ & =& {\displaystyle [\frac{(\beta -\alpha {)}^3}{3}-0]-[\frac{(\beta -\alpha )(\beta -\alpha {)}^2}{2}-0]}\\ & =& {\displaystyle \frac{(\beta -\alpha {)}^3}{3}-\frac{(\beta -\alpha {)}^3}{2}}\\ & =& {\displaystyle \frac{2(\beta -\alpha {)}^3-3(\beta -\alpha {)}^3}{6}}\\ & =& {\displaystyle \frac{-(\beta -\alpha {)}^3}{6}}\end{array}$$ これは第一種オイラー積分の一つの形であり，$\frac{1}{6}$公式と呼ばることがある式である．