- Integrate $f(x) = 2x^3$ between 1 and 5.
- With two equal intervals, use the midpoint and trapezoid and to approximate $f(x)$. Show that the midpoint rule is roughly twice as accurate as the trapezoid rule.
- Demonstrate that Simpson's rule integrates cubics exactly by calculating it for $f(x)$ with two intervals.
- Show that Simpson's Composite Rule and Weddle's rule both produce the same result.
- Show that the two forms of Simpson's rule are equivalent:
\begin{equation}
S = \frac{2}{3}M + \frac{1}{3}T
\end{equation}\begin{equation}
S = \frac{h}{6}\left ( f \left ( a\right ) + 4f \left ( c\right ) + f\left ( b\right )\right )
\end{equation}