We now introduce the second primary tool of calculus, the integral. While the integral seems, at first glance, to be completely unrelated to the derivative, we will see later on that the two are intimately connected. This relation between the derivative and integral is so important that its formal statement is referred to as the Fundamental Theorem of Calculus.
The definite integral arises in the study of the area of a certain type of region in the plane: regions bounded above by the graph of a function f (x), below by the x-axis, and on the sides by two vertical lines corresponding to the equations x = a and x = b. Indeed, for two real numbers a < b and a function f (x) defined on the interval [a, b], we denote by R(f, a, b) the region consisting of points (x, y) with a≤x≤b and with y between 0 and f (x).
We want to compute the area of such a region. However, while we know what we mean by the area of, say, a rectangle (base times height), its not quite clear what we mean by the area of an arbitrary region. Hence we need to come up with a way of precisely defining area, at least for regions of the form R(f, a, b).
We choose to do this by approximating the region with a number of thin rectangular regions and calculating the total area of these rectangles, using the familiar formula of base times height. This sum of areas of rectangles is called a Riemann sum. As we let the width of the rectangles get smaller and smaller, the approximation of the area of the region will become more and more accurate. The limit of these approximate areas as the width of the rectangles approaches 0, if it exists, is what we define to be the area of the actual region. The definite integral of a function from one x-value to another is the name given to this limit of approximate areas.