Do you know the geometric idea behind Riemann series? Basically the point is adding areas of rectangles, of height $\displaystyle f(c_{i})$ and width $\displaystyle \Delta x_{i}$. If you have the following sequence of points
$\displaystyle x_{0},x_{1},x_{2},x_{3},x_{4}$
you will have the following sequence of widths
$\displaystyle \Delta x_{1}=x_{1}-x_{0},\Delta x_{2}=x_{2}-x_{1},\Delta x_{3}=x_{2}-x_{1},\Delta x_{4}=x_{3}-x_{2}$
and the following sequence of mid points $\displaystyle c_{i}$:
$\displaystyle c_{1}=\frac{x_{1}-x_{0}}{2},c_{2}=\frac{x_{2}-x_{2}}{2},c_{3}=\frac{x_{2}-x_{1}}{2},c_{4}=\frac{x_{4}-x_{3}}{2}$