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More about Integration E-7

In many cases you will be familiar enough with the function you want to integrate

that you will know whether the function has any quick wiggles relative to the interval

of integration. If you're not familiar with the function, and you suspect that it may

cause problems, you can quickly plot a few points by evaluating the function using

the equation or program you wrote for that purpose.

If, for any reason, after obtaining an approximation to an integral, you suspect its

validity, there's a simple procedure to verify it: subdivide the interval of integration

into two or more adjacent subintervals, integrate the function over each subinterval,

then add the resulting approximations. This causes the function to be sampled at a

brand new set of sample points, thereby more likely revealing any previously hidden

spikes. If the initial approximation was valid, it will equal the sum of the

approximations over the subintervals.

Conditions That Prolong Calculation Time

In the preceding example, the algorithm gave an incorrect answer because it never

detected the spike in the function. This happened because the variation in the

function was too quick relative to the width of the interval of integration. If the width

of the interval were smaller, you would get the correct answer; but it would take a

very long time if the interval were still too wide.

Consider an integral where the interval of integration is wide enough to require

excessive calculation time, but not so wide that it would be calculated incorrectly.

Note that because f(x) = xe

–x

approaches zero very quickly as x approaches ∞, the

contribution to the integral of the function at large values of x is negligible.

Therefore, you can evaluate the integral by replacing ∞, the upper limit of

integration, by a number not so large as 10

499

— say 10

Rerun the previous integration problem with this new limit of integration:

Keys: Display: Description:





_ New upper limit.



 Selects Equation mode; displays the

equation.

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