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.