Hippocrates of Chios
Scholium: the Integral
At the other end
of the trajectory of ideas that began with the area problem in ancient
Greece is the modern calculus concept of the integral. Loosely speaking,
the integral is an area calculator. Given a line or curve in the xy-plane
with equation y = f(x) (here, f(x) represents
any algebraic expression in x), there is a region bounded by this
curve and the x-axis. If we assume for the moment that the curve
lies above the axis, then the region has this curve as top edge and the
axis as bottom edge. If we specify two numbers a < b on
the x-axis and erect the vertical lines perpendicular to the axis
at these points (x = a and x = b are the equations of these
vertical lines), these lines can serve as left and right boundaries of
the region.
We therefore have a region bounded
on three sides by straight lines and on the fourth by a curve whose equation
we know. We use the notation
to represent the area of this region.
This is read "the integral of f(x) between a
and b." The integral symbol was first used by Leibniz
in 1686. The values placed as subscript and superscript next to the integral
symbol are respectively called the lower limit and upper limit
of integration; the expression that follows is called the integrand;
and the symbol dx to the right of the integrand is called a differential.
The notation gives all the necessary information for describing the region
whose area we wish to determine: the integral symbol indicates that an
area above the x-axis is to be calculated, the limits of integration
indicate the left- and right-side bounds of the region, the integrand determines
the upper boundary of the region, and the differential points to the x
as the underlying input variable.
For example, the integral
represents the area
of the trapezoid bounded by the x-axis, the vertical lines x
= 2 and x = 5, and the line y = 2x + 1. The reader
is urged to sketch a graph of this region (perhaps with the aid of a graphics
calculator) to see which trapezoid we are considering here. Notice that
the same region is represented by the integral
since the only thing
that has changed is the name of the input variable.
Since there is a large class of plane figures whose areas we can calculate
from basic geometry, there is then a large class of integrals that we can
compute, even without the benefit of sophisticated machinery from calculus.
For instance,
since the integral
asks for the area of a rectangle (the graph of y = 2 is a horizontal
line) 4 units wide by 2 units tall. In general, if k is any constant,
We also have
since the region we
are considering here is a triangle with base 8 units long and height 3
¥
8 = 24 units tall, whence area = (1/2)(8 ¥
24) = 96.
Other simple figures can be represented by not-so-simple looking integrals.
For instance,
asks for the area
of a trapezoid, and
the area of a semicircle.
(Check these out with a graphics calculator!)
As we develop more tools for handling the area problem, we will be able
to extend the class of integrals we can compute even further. The exercises
for this chapter include some explorations of these ideas.
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last
modified 8/28/02
Copyright (c) 2000.
Daniel E. Otero