Numerical Calculus Project
Part II: Integration
Context:
Your next project will be all about numerical calculus. This project will be split into three parts.
The first two will be preliminary tasks where you will become familiar with numerical differentiation and
integration and write and test drive methods to compute numerical derivatives and integrals. The final part of
the project will use the functions from the first two parts and bring them together with an interface and
analysis. This is the second part. This part of the project requires you to use skills up to Exam 2 and what you
have learned about data files.
Theory
Recall from calculus the concept of the integral. In this project, we'll be using the computer to help us
compute definite integrals of functions, ultimately with the goal of finding the area under a curve between
given bounds. As you hopefully learned in calculus, we can investigate mathematics via the rule of three 
analytically, graphically, and numerically. There are many cases where computing integrals analytically is
very challenging, tedious, or impossible, so we'll once again use numerical techniques to investigate them.
We call this numerical integration process
quadrature
. Ultimately, the same idea is at the heart of all of our
quadrature methods: take the area under the curve and divide and conquer, i.e. break it up into smaller areas
we can compute or approximate more easily and add them up to get an approximation of the area we want.
You likely saw some of this material in your Calculus I experience. We'll begin there, with Riemann sums.
Let's suppose we're aiming to find the definite integral of
f
(
x
) between the bounds of
x = a
and
x = b
.
There are two basic starting strategies:
•
Left Sum:
Divide the interval [
a
,
b
] up into
n
parts, effectively dividing the area under
f
(
x
) into
n
parts as well. Use rectangles of width (
ba
)/
n
to approximate the area. The height of each rectangle
should be
f
(
x
L
), where
x
L
is the left endpoint of the interval.
•
Right Sum:
Similarly, divide the interval [
a
,
b
] up into
n
parts, effectively dividing the area under
f
(
x
) into
n
parts as well. Use rectangles of width (
ba
)/
n
to approximate the area. This time, however,
use
f
(
x
R
), where
x
R
is the right endpoint of the interval, as the height of each rectangle.
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 Fall '10
 Numerical Analysis, Riemann sum, quadrature technique

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