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Integration of a function is familiar from introductory calculus courses,
where integration is usually introduced in two ways. The first is as the
inverse process to differentiation, as the so-called anti-derivative of
a function. Thus if we have

then we invert this to say that *f*(*x*) is the integral of *g*(*x*). While
this is not a particularly satisfying definition of integration, it does
reflect the usual methods we use to evaluate integrals in closed form,
when this is possible. As a very simple example, one simply says that
the integral of
is ,
because the
derivative of
with respect to
is .
Of course, there is an arbitrary additive constant to be added to the
integral described in this way. In the notation of integral calculus,
we would write:

where C is the arbitrary constant with respect to *x*. *g*(*t*) is called the
integrand, and *t* is the variable of integration. It will be important to
note that *t* is a ``dummy'' variable of integration, and that the integral
itself is not a function of this ``dummy'' variable, but of the limiting
values it takes. Consider the rather simple function:

*g*(*x*) = *A x*^{2} + *B x* + *C*,

where *A*, *B*, and *C* are constant with respect of *x*. To find the
purely symbolic integral of *g*(*x*), we would evaluate:

where *D* is a constant of integration. Integrals in this symbolic form
are called *indefinite* integrals, since they do not yield any definite
result until evaluated between actual limits, so that the constant of
integration, *D* in this example, takes on a specific value. Thus a
*definite* integral obtainable from the indefinite form would be:

which would yield the value:

The [ ... ] expression means to evaluate the contents for the *upper*
limit (
in this example) and subtract from it the value of the contents
evaluated at the *lower* limit (
in this example). Thus the
quantity
is not a function of *x* at all, but definitely
a function of its two arguments,
and .
In many applications of calculus to problems that arise in science and
engineering, the integrals that are involved in calculating the relevant
quantities are not easily evaluated in symbolic form. By this we mean that
it is not easily possible to find the appropriate function, whose
derivative is the integrand of the integral.

** Next:** Area Under the Curve
** Up:** Numerical Integration
** Previous:** Numerical Integration
*Charles Dyer*

*2002-04-24*