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The left hand rule, right hand rule, midpoint rule, and composite trapezoidal rule, all with $n=6$ , will be used in turn to estimate the Riemann integral $\displaystyle \int\limits_{-1}^2 x^2 \, dx$ .
Since $n=6$ , $\displaystyle \frac{b-a}{n}=\frac{2-(-1)}{6}=\frac{3}{6}=\frac{1}{2}$ .
- Left hand rule:
- Right hand rule:
- Midpoint rule:
- Composite trapezoidal rule:
For comparison purposes, the Riemann integral will also be computed:
As expected, of the three estimates, the ones obtained from the midpoint rule and the composite trapezoidal rule are closest to the actual value of the Riemann integral. Their errors are $\displaystyle \frac{1}{16}$ and $\displaystyle \frac{1}{8}$ , respectively. It may seem odd that the midpoint rule should be closer to the actual value of the Riemann integral even though, in the graph for the composite trapezoidal rule, the approximating trapezoids are barely distinguishable from the parabola. Actually though, this is to be expected, as the maximum error for the midpoint rule is half of the maximum error for the composite trapezoidal rule.
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