Computing integrals is much more complex than computing derivatives. This is the reason that before the era of computers and calculators there were only two practical ways of doing it. The first one, which is still taught in most calculus courses, is to find a formula for the anti-derivative of the function whose integral we seek. It is a very useful technique when it can be applied. But not all functions have anti-derivatives and even when they do, then often there are no "nice" formulas for them that can be used. The second method that was used for some especially important functions (for example, "elliptic functions") was to prepare tables of values (which was a costly and slow process) and later use them in the same way as other tables (logarithmic, trigonometric, statistical) were used.

At present the computing power of calculators is big enough so that numerical techniques that require hundreds or thousands of operations have become available. The method that is actually used was developed by Gauss and improved and modified by other mathematicians.

The basic idea is simple. An integral of a function f(x) relative to x changing from a to b, can be approximated by a sum of f(x)*dx where x is computed for properly selected values of x, and where values of dx are the sizes of intervals from which x is chosen.

Technically this means that we need to choose n values x_{1},
x_{2}, ..., x_{n}, between a and b;
choose sizes of intervals s_{1}, s_{2}, ..., s_{n}; compute the values f(x_{1}), f(x_{2}),
..., f(x_{n}); and finally compute the sum of
products,

f(x_{1})*s_{1} + f(x_{2})*s_{2}
+ ... + f(x_{n})*s_{n}

In general, bigger values of n provide better accuracy, but they also make the computation slower. With a "naive" choice of x's, for example, equally spaced, getting good accuracy would require too much time to be useful.

But Gauss showed a very smart method of choosing the x's, based on a polynomial approximation of the function f(x),that provides very good accuracy for already small values of n. This is the method that is actually implemented on the TI-83/84 calculators.

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