The definite integral corresponds to the area under the graph of a function, above the x-axis and between two vertical lines called the limits of integration. Consider approximating the area under the graph of a function f(x) by drawing a series of rectangles, and summing their areas to arrive at the total area, A(x) (see Figure 2). The height of each rectangle is the value of the function at x, namely f(x). The width of each rectangle is Δx = (b-a)/n, where n is the number of rectangles chosen. If we wish to know the area between x=a and x=b, then the area is given by the sum
(the Greek letter Σ (sigma) is used to indicate that the n products f(xk) Δx corresponding to the n rectangles are to be summed). In the limit that n approaches infinity, Δx approaches 0, and the sum is exactly equal to the are. Since Δx approaches 0, it represents an infinitesimal change in the variable x, so the same notation used in defining the derivative is used to replace Δx with d. The product f(x) Δx becomes f(x)dx, and corresponds to an infinitesimal area, dA(x). The total area, then, is the sum of an infinite number of an infinite number of infinitesimal areas. Thus, the area A(x) between a and b is equal to the integral of f(x)dx, written,
The limits included above and below the integral sign indicate that the indefinite integral of f(x)d is to be evaluated at a and b and the values subtracted, that is,
This is interpreted as the area under the curve to the left of b minus the area under the curve to the left of a. Comparing this with the form of the indefinite integral we see that a function f(x) is the derivative of its "Area function," the constant C being evaluated by use of boundary conditions, namely the values a and b. There are many applications of definite integrals, among the most common are the determination of areas and volumes of revolution.
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