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What is Integration?

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

\begin{displaymath}
\frac{df}{dx} = g(x)
\end{displaymath}

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 $\cos(\theta)$ is $\sin(\theta)$, because the derivative of $\sin(\theta)$ with respect to $\theta$ is $\cos(\theta)$. 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:

\begin{displaymath}
\int^x g(t) dt = f(x) + C,
\end{displaymath}

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 x2 + 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:

\begin{displaymath}
\int^x (A t^2 + B t + C) dt = \frac{A}{3} x^3 + \frac{B}{2} x^2 + C x + D,
\end{displaymath}

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:

\begin{displaymath}
F(\alpha,\beta) = \int_{\alpha}^{\beta} (A t^2 + B t + C) dt
\end{displaymath}

which would yield the value:

\begin{displaymath}
F(\alpha,\beta) = \left [ \frac{A}{3} x^3 + \frac{B}{2} x^2 + C x \right ]
_{x = \alpha}^{x = \beta}.
\end{displaymath}

The [ ... ] expression means to evaluate the contents for the upper limit ($\beta$ in this example) and subtract from it the value of the contents evaluated at the lower limit ($\alpha$ in this example). Thus the quantity $F(\alpha,\beta)$ is not a function of x at all, but definitely a function of its two arguments, $\alpha$ and $\beta$.

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 up previous
Next: Area Under the Curve Up: Numerical Integration Previous: Numerical Integration
Charles Dyer
2002-04-24