Midpoint Riemann sum for sin(x) on [0, pi] with n=6

The width of each subinterval is $\Delta x = \frac{\pi - 0}{6} = \frac{\pi}{6}$. The subintervals are $[0,\frac{\pi}{6}], [\frac{\pi}{6},\frac{\pi}{3}], [\frac{\pi}{3},\frac{\pi}{2}], [\frac{\pi}{2},\frac{2\pi}{3}], [\frac{2\pi}{3},\frac{5\pi}{6}], [\frac{5\pi}{6},\pi]$, so the midpoints are $\frac{\pi}{12}, \frac{\pi}{4}, \frac{5\pi}{12}, \frac{7\pi}{12}, \frac{3\pi}{4}, \frac{11\pi}{12}$.

Evaluating $\sin(x)$ at these midpoints gives approximately $0.2588, 0.7071, 0.9659, 0.9659, 0.7071, 0.2588$. Summing these values gives about $3.8637$, and multiplying by $\Delta x = \frac{\pi}{6} \approx 0.5236$ gives $M_6 \approx 2.0233$.