HP 33s Conditions That Prolong Calculation Time, E–7, 0 ‘a3, _, %º% 1.%2, New upper limit, More about Integration

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–xapproaches 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 10499 — say 103.

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

Keys:Display:Description:

0 ‘a3	_
H	%º%1.%2
X	!!
	≥ /
	)

New upper limit.

Selects Equation mode; displays the equation.

Integral. (The calculation takes a minute or two.)

More about Integration E–7