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'trapezoidal rule'
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| Title of object: |
trapezoidal rule |
| Canonical Name: |
TrapezoidalRule |
| Type: |
Definition |
| Created on: |
2003-06-02 23:09:16 |
| Modified on: |
2007-06-10 10:02:17 |
| Classification: |
msc:41A05, msc:41A55 |
| Synonyms: |
trapezoidal rule=trapezoid rule
trapezoidal rule=trapezium rule |
Revision comment (for changes between this and next version):
| Changes for correction #12304 ('trapezoid'). |
Preamble:
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Content:
\PMlinkescapeword{proposition}
\textbf{Definition}\\
The \emph{trapezoidal rule} is a method for approximating a definite
integral by evaluating the integrand at two points. The
formal rule is given by
\[
\int_{a}^{b}f(x)\,dx\;\approx\;\frac{h}{2}\left[f(a)+f(b)\right]
\]
where $h=b-a$.
The trapezoidal rule is the first Newton-Cotes quadrature formula.
It has degree of precision 1. This means it is exact for
polynomials of degree less than or equal to one. We can see this
with a \PMlinkescapetext{simple} example.
\textbf{Proposition:}\\
If $f$ is Riemann integrable on $[a,b]$, $|f''(x)| \le M$ for all $x \in [a,b]$ then
$$\left| \int_a^b f(x) \, dx - \frac{h}{2}\left[f(a)+f(b)\right]
\right| \le \frac{M(b-a)^3}{12}.$$
\textbf{Example:}
Using the fundamental theorem of the calculus shows
\[
\int_{0}^{1}x\,dx =1/2.
\]
In this case the trapezoidal rule gives the exact value,
\[
\int_{0}^{1}x\,dx \;\approx\;\frac{1}{2}[f(0)+f(1)]=1/2.
\]
It is important to note that most calculus books give the wrong
definition of the trapezoidal rule. Typically they define a
composite trapezoidal rule which uses the trapezoidal rule on a
specified number of subintervals. Also note the trapezoidal rule
can be derived by integrating a linear interpolation or using the
method of undetermined coefficients. The latter is probably a bit
easier. |
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