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More about Integration

E–3

File name 33s-English-Manual-040130-Publication(Edition 2).doc Page : 388

Printed Date : 2004/1/30 Size : 13.7 x 21.2 cm

f (x)

With this number of sample points, the algorithm will calculate the same

approximation for the integral of any of the functions shown. The actual integrals

of the functions shown with solid blue and black lines are about the same, so the

approximation will be fairly accurate if

f(x) is one of these functions. However, the

actual integral of the function shown with a dashed line is quite different from those

of the others, so the current approximation will be rather inaccurate if

f(x) is this

function.

The algorithm comes to know the general behavior of the function by sampling the

function at more and more points. If a fluctuation of the function in one region is

not unlike the behavior over the rest of the interval of integration, at some iteration

the algorithm will likely detect the fluctuation. When this happens, the number of

sample points is increased until successive iterations yield approximations that

take into account the presence of the most rapid,

but characteristic, fluctuations.

For example, consider the approximation of

∫

∞

−

dxxe

Since you're evaluating this integral numerically, you might think that you should

represent the upper limit of integration as 10

499

, which is virtually the largest

number you can key into the calculator.

Try it and see what happens. Enter the function

f(x) = xe

–x

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