انتگرال ریمان–استیلتیس

${\displaystyle \Delta \alpha _{i}=\alpha (x_{i})-\alpha (x_{i-1})}$

${\displaystyle U(P,f,\alpha )=\sum _{i=1}^{n}M_{i}\Delta \alpha _{i}}$

${\displaystyle L(P,f,\alpha )=\sum _{i=1}^{n}m_{i}\Delta \alpha _{i}}$

${\displaystyle M_{i}=M_{i}(f)=sup\{f(x)|x_{i-1}\leq x\leq x_{i}\}}$

${\displaystyle m_{i}=m_{i}(f)=inf\{f(x)|x_{i-1}\leq x\leq x_{i}\}}$

${\displaystyle \int _{a}^{-_{b}}f\,d\alpha =\inf _{P}U(P,f,\alpha )}$

${\displaystyle \int _{-_{a}}^{b}f\,d\alpha =\sup _{P}L(P,f,\alpha )}$