We wanted to follow up with a post that contains a dump of our data. This includes, essentially, our percent error from the expected value of the given test functions:

- $\cos{x}$ over [0, $\pi{}$]
- $2x + 1$ over [0, 1]
- $4-x^2$ over [0, 2]
- $5x^3 – 6x^2 + 0.3x$ over [-1, 3]
- $x^3$ over [-1, 3]
- $x^3 -27x^2 + 8x$ over [0, 3]

We tested 5 different deltas (rectangle widths), $dx$, namely, $0.1$, $0.01$, $0.001$, $0.0001$, $0.00001$. But we are not going to put tables for each method **and** each delta; it’s just too much. However, we will do the first delta ($0.1$) and the last delta ($0.00001$).

### Summary of Methods for $\cos{x}$ over [0, $\pi{}$]

Method |
Delta |
Percent Error |
---|---|---|

Trapezoidal | $0.100000$ | -0.33364 |

Trapezoidal | $0.000010$ | -0.00000 |

Midpoint | $0.100000$ | -0.20893 |

Midpoint | $0.000010$ | -0.00000 |

Simpsons | $0.100000$ | 0.05475 |

Simpsons | $0.000010$ | 0.00000 |

Left Rectangle | $0.100000$ | -4.97995 |

Left Rectangle | $0.000010$ | -0.00050 |

Right Rectangle | $0.100000$ | 4.31267 |

Right Rectangle | $0.000010$ | 0.00050 |

### Summary of Methods for $2x + 1$ over [0, 1]

Method |
Delta |
Percent Error |
---|---|---|

Trapezoidal | $0.100000$ | -14.50000 |

Trapezoidal | $0.000010$ | -0.00150 |

Midpoint | $0.100000$ | -14.50000 |

Midpoint | $0.000010$ | -0.00150 |

Simpsons | $0.100000$ | 0.00000 |

Simpsons | $0.000010$ | 0.00000 |

Left Rectangle | $0.100000$ | -10.00000 |

Left Rectangle | $0.000010$ | -0.00100 |

Right Rectangle | $0.100000$ | -19.00000 |

Right Rectangle | $0.000010$ | -0.00200 |

### Summary of Methods for $4-x^2$ over [0, 2]

Method |
Delta |
Percent Error |
---|---|---|

Trapezoidal | $0.100000$ | -0.42813 |

Trapezoidal | $0.000010$ | -0.00000 |

Midpoint | $0.100000$ | -0.33906 |

Midpoint | $0.000010$ | -0.00000 |

Simpsons | $0.100000$ | 0.00000 |

Simpsons | $0.000010$ | -0.00000 |

Left Rectangle | $0.100000$ | -3.81250 |

Left Rectangle | $0.000010$ | -0.00038 |

Right Rectangle | $0.100000$ | 2.95625 |

Right Rectangle | $0.000010$ | 0.00037 |

### Summary of Methods for $5x^3 – 6x + 0.3x$ over [-1, 3]

Method |
Delta |
Percent Error |
---|---|---|

Trapezoidal | $0.100000$ | -16.93086 |

Trapezoidal | $0.000010$ | -0.00181 |

Midpoint | $0.100000$ | -17.10882 |

Midpoint | $0.000010$ | -0.00181 |

Simpsons | $0.100000$ | -0.00000 |

Simpsons | $0.000010$ | -0.00000 |

Left Rectangle | $0.100000$ | -7.67699 |

Left Rectangle | $0.000010$ | -0.00078 |

Right Rectangle | $0.100000$ | -26.18473 |

Right Rectangle | $0.000010$ | -0.00284 |

### Summary of Methods for $x^3$ over [-1, 3]

Method |
Delta |
Percent Error |
---|---|---|

Trapezoidal | $0.100000$ | -14.87537 |

Trapezoidal | $0.000010$ | -2.44034 |

Midpoint | $0.100000$ | -15.01091 |

Midpoint | $0.000010$ | -2.44034 |

Simpsons | $0.100000$ | -2.43902 |

Simpsons | $0.000010$ | -2.43902 |

Left Rectangle | $0.100000$ | -8.68293 |

Left Rectangle | $0.000010$ | -2.43966 |

Right Rectangle | $0.100000$ | -21.06780 |

Right Rectangle | $0.000010$ | -2.44102 |

### Summary of Methods for $x^3 – 27x^2 + 8x$ over [0, 3]

Method |
Delta |
Percent Error |
---|---|---|

Trapezoidal | $0.100000$ | -9.88570 |

Trapezoidal | $0.000010$ | -0.00103 |

Midpoint | $0.100000$ | -9.97363 |

Midpoint | $0.000010$ | -0.00103 |

Simpsons | $0.100000$ | 0.00000 |

Simpsons | $0.000010$ | -0.00000 |

Left Rectangle | $0.100000$ | -5.08032 |

Left Rectangle | $0.000010$ | -0.00051 |

Right Rectangle | $0.100000$ | -14.69108 |

Right Rectangle | $0.000010$ | -0.00154 |

Of course, we must mention that there is some rounding in the percent errors. Simpsons, Midpoint, and Trapezoidal methods are not perfect.

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