Find the Riemann sum … for … in Remark 1, when …

If $v(x)$ is increasing for $a⩽x⩽b$, the difference $S-s$ is the area of the $\underline{}$ rectangle minus the area of the $\underline{}$ rectangle. Those areas approach zero. So every increasing function on $[a,b]$ is Riemann integrable.

Find the Riemann sum $S^{\ast }$ for $V(x)$ in Remark 1, when $\Delta x=1/n$ and each $x_k^{\ast }$ is the midpoint. This $S^{\ast }$ is well-behaved but still $V(x)$ is not Riemann integrable.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §5.5 The Definite Integral, #26