|
The definite integral with respect to $x$ of some function $f(x)$ over the compact interval $[a,b]$ with $a<b$ , the interval of integration, is defined to be the ``area under the graph of $f(x)$ with respect to $x$ '' (if $f(x)$ is negative, then you have a negative area). The
numbers $a$ and $b$ are called lower and upper limit respectively. It is written as: $$ \int_a^bf(x) \ dx .$$ One way to find the value of the integral is to take a limit of an approximation technique as the precision increases to infinity.
For example, use a Riemann sum which approximates the area by dividing it into $n$ intervals of equal widths, and then calculating the area of rectangles with the width of the interval and height dependent on the function's value in the interval. Let $R_n$ be this approximation, which can be written as $$ R_n = \sum_{i=1}^{n} f(x_i^*) \Delta x ,$$ where $x_i^*$ is some $x$ inside the $i^{\rm th}$ interval. This process is illustrated by figure
1.
Figure 1: The area under the graph approximated by rectangles
|
|
Then, the integral would be $$ \int_a^bf(x) \ dx = \lim_{n \to \infty} R_n = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x .$$ This limit does not necessarily exist for every function $f$ and it may depend on the particular choice of the $x_i^*$ . If all those limits coincide and are finite, then the integral exists. This is true in particular for continuous $f$ .
Furthermore we define $$\int_b^af(x)\ dx=-\int_a^bf(x)\ dx.$$
We can use this definition to arrive at some important properties of definite integrals ($a$ , $b$ , $c$ are constant with respect to $x$ ): \begin{eqnarray*} \int_a^b(f(x) + g(x)) \ dx & = & \int_a^bf(x)\ dx + \int_a^bg(x)\ dx; \\ \int_a^b(f(x) - g(x)) \ dx & = & \int_a^bf(x)\ dx - \int_a^bg(x)\ dx ;\\ \int_a^bf(x) \ dx & = & \int_a^cf(x)\ dx + \int_c^bf(x)\ dx ;\\ \int_a^bcf(x) \ dx & = & c\int_a^bf(x)\ dx. \end{eqnarray*} There are other generalizations about integrals, but many require the fundamental theorem of calculus.
| 
