Math 172: Integral Calculus
Prof. Thomas Pietraho
Fall, 2007
Lab 1: Approximation Errors
Part 1
1
Motivation
Many functions do not have an elementary antiderivative, and to evaluate a definite integral of such
a function, we will need to resort to methods of approximation.
The goal of this lab is to examine the
approximation of a definite integral using its Riemann sums. Let’s first examine the integral
S
=
Z
π
0
e

x
2
dx.
We are most interested in finding out the effect of increasing the number of intervals used in the sum has on
the accuracy of the approximation.
2
Left Riemann Sums
Let’s first get some data for the integral above.
Mathematica
can find the value of
S
exactly, and using
the
Joy of Mathematica
menus, it is possible to find the left Riemann sums for different numbers of intervals.
1. For starters, first determine the numerical value of
S
using
Mathematica
.
2. Let’s use the left end point method to estimate
S
using 2, 4, 8, 16, 32, 64, and 128 subintervals.
Joy
facilitates this computation through the
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 Fall '08
 Pietraho
 Calculus, Approximation, Derivative, Logarithm, Riemann sum

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