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'definite integral'
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| Title of object: |
definite integral |
| Canonical Name: |
DefiniteIntegral |
| Type: |
Definition |
| Created on: |
2002-02-02 01:49:37 |
| Modified on: |
2006-01-31 03:52:18 |
| Classification: |
msc:26A06 |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx} |
Content:
The \emph{definite integral} with respect to $x$ of some function $f(x)$ over the closed interval $[a,b]$ is
defined to be the ``area under the graph of $f(x)$ with respect to $x$'' (if $f(x)$ is negative, then you have a negative area). It is written as:
$$ \int_a^bf(x) \ dx $$
One way to find the value of the integral is to take a limit of an approximation technique
as the precision increases to infinity.
For example, use a Riemann sum which approximates
the area by dividing it into $n$ intervals of equal widths, and then calculating the area
of rectangles with the width of the interval and height dependent on the function's value in the interval.
Let $R_n$ be this approximation, which can be written as
$$ R_n = \sum_{i=1}^{n} f(x_i^*) \Delta x $$
where $x_i^*$ is some $x$ inside the $i^{\rm th}$ interval. This process is illustrated by figure \ref{fig:bars}.
\begin{figure}[htbp]
\begin{centering}
\includegraphics[angle=270,scale=0.5]{definite_integral.ps}
\caption{The area under the graph approximated by rectangles}\label{fig:bars}
\end{centering}
\end{figure}
Then, the integral would be
$$ \int_a^bf(x) \ dx = \lim_{n \to \infty} R_n =
\lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x $$
We can use this definition to arrive at some important properties of definite integrals
($a$, $b$, $c$ are constant with respect to $x$):
\begin{eqnarray*}
\int_a^b(f(x) + g(x)) \ dx & = & \int_a^bf(x)\ dx + \int_a^bg(x)\ dx \\
\int_a^b(f(x) - g(x)) \ dx & = & \int_a^bf(x)\ dx - \int_a^bg(x)\ dx \\
\int_a^bf(x) \ dx & = & - \int_b^af(x)\ dx \\
\int_a^bf(x) \ dx & = & \int_a^cf(x)\ dx + \int_c^bf(x)\ dx \\
\int_a^bcf(x) \ dx & = & c\int_a^bf(x)\ dx
\end{eqnarray*}
There are other generalizations about integrals, but many require the fundamental theorem of calculus. |
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