12-06-2013, 03:34 PM
The Denite Integral
The Denite.pdf (Size: 550.7 KB / Downloads: 305)
Introduction
The Denite Integral has wide ranging applications in mathematics,
the physical sciences and engineering.
The theory and application of statistics, for example, depends heavily
on the denite integral; through statistics, many traditionally nonmathematical
disciplines have become heavily dependent on mathematical
ideas. Economics, sociology, psychology, political science, geology,
geography, and many others professional elds utilize calculus
concepts.
Unlike the Indenite Integral, which is a function, the Denite Integral
is a numerical value. As we shall see, on rst inspection, there seems
to be no relation between these two mathematical objects, but as the
theory unfolds, their relationship will be revealed.
The Denite Integral, as has been stated already, has wide-ranging
application; however the problem is the diverse backgrounds of students
taking Calculus. Some may know a lot of physics, while others
may have good knowledge of electrical circuits. In a Calculus course,
no general background in the sciences is assumed, as a result, the
applications that tend to be presented are of two types:
1. Geometric Applications: All students have a general background
in geometry. Drawing curves should be your fortee. Consequently,
many of the applications to the Denite Integral seen in traditional
Calculus courses are geometric: Calculation of Area, Calculation of
Volume, Calculation of Surface Area, and calculation of Arc Length.
These applications are good in the sense that they allow the student
to see some useful applications, but more importantly, the students
sees the process of constructing the application.
2. Physical Applications: There are some physical applications to
the Denite Integral usually seen in a course on Calculus. Because
of the diverse backgrounds of students, these applications tend to be
easily accessed by everyone: The physical notions of work, hydrostatic
pressure, mass, and center of mass.
A Little Problem with Area
We begin by motivating the construction of the Denite Integral with
a particular application | to that of the Area problem.
Like, what is it? Throughout your school experiences, the notion of
area has been a fundamental one. Area is a concept whose meaning has
been built up through a series of denitions and deductions through
the years of your education. Area has been inculcated into you through
these years until it has become second nature to you.
Passing to the Limit
Now, here's were the advanced notions of Calculus come in. Up to
this point, it has been all notation and algebra. We must make the
leap from the approximate to the exact.