Today: 7.7 - approximating integrals
Friday: Third QUIZ and 7.8 - improper integrals |

Problem: Compute

Today we will revisit Riemann sums in the context of finding numerical approximations to integrals, which we might not be able to compute exactly. Recall that if then

For example, we could use Riemann sums to approximate
,
say using *left endpoints*. This gives the approximation:

left endpoints

right endpoints

midpoints

where
. The midpoint is
typically (but not always) much better than the left or right endpoint
approximations.
Yet another possibility is the *trapezoid approximation*,
which is

Many functions have no elementary antiderivatives:

Some of these functions are extremly important. For example, the integrals are extremely important in probability, even though there is no simple formula for the antiderivative.

If you are doing scientific research you might spend months tediously computing values of some function , for which no formula is known.

- Trapezoid with
- Midpoint with
- Simpson's with with

The following is a table of the values of at for .

0 | |

Maxima gives and Mathematica gives .

Note that Simpsons's is the best; it better be, since we worked the hardest to get it!

Method | Error |

0.101573 | |

0.056458 | |

0.005917 | |

0.022558 | |

0.003575 |

William Stein 2006-03-15