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The trapezium rule states that you can approximate the area under a
curve by a trapezium, as shown in figure C.4. The area is
therefore calculated like this:
Figure C.4:
A straight line is used to approximate the curve f (x)
between x_{1} and x_{2}. This forms a trapezium (shaded area) which can be
used to calculate approximately the area under
the curve between these bounds.

f (
x)
dx =
hf_{1} +
f_{2} +
O(
h^{3}f'')
Where the notation f_{1} is short hand for f (x_{1}).
The error due to the approximation,
O(h^{3}f''), indicates that this
method is exact for polynomials up to degree 1, i.e. straight lines
f (x) = ax + b, because for such functions the second derivative
f'' = 0. It is clearly an approximation for higher order polynomials
(x^{2},x^{3}, etc.) and other functions. The error will also be small
if h is small.
The area of a trapezium is its average height x its width, i.e.
(f_{1} + f_{2}).
Next: Extended trapezium rule
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RHUL Dept. of Physics