Trapezoidal endpoint correction for a right Riemann sum ($R_N-T_N=\frac{h}{2}(f(b)-f(a))$)

On a uniform partition of $[a,b]$ with width $h$, the right Riemann sum and the composite trapezoidal rule have equal weights at every interior mesh point. They differ only by adding half a final-endpoint weight and removing half an initial-endpoint weight. This proves the displayed identity. For continuous $f$, both converge to its definite integral; their finite endpoint correction need not be confused with the smaller trapezoidal discretization error for a smooth integrand.