III.

Integral Calculus

Let y = f(x) be a function defined for all x’s in the interval [a,b], that is, the set of x’s from x = a to x = b, including a and b, where a<b (suitable modifications can be made in the definitions to follow for more restricted ranges or domains). Let x₀, x₁, ..., x_n be a sequence of values of x such that a = x₀<x₁<x₂<...<x_n - 1 <x_n = b, and let h₁ = x₁ - x₀, h₂ = x₂ - x₁, ..., h_n = x_n - x_{n - 1}, in brief, h_i = x_i - x_{i - 1}, where i = 1, 2, ..., n. The x’s form a partition of the interval [a, b]; an h with a value not exceeded by any other h is called the norm of the partition. Let n values of x, for example, X₁, X₂, ..., X_n, be chosen so that x_{i - 1}<X_i<x_i, where i = 1, 2, ..., n. The sum of the area of the rectangles is given by

f(X1)h1 + f(X2)h2 + .... + f(Xn)hn

usually abbreviated to

(Σ is the Greek capital letter sigma.) Aside from the given function f(x) and the given a and b, the value of the sum clearly depends on n and on the choices of the x_i’s and X_i’s. In particular, if, after the x_i’s are chosen, the X_i’s are chosen so that f(X_i), for each i, is a maximum in the interval [x_{i - 1}, x_i] (that is, no ordinate from x_{i - 1} to x_i exceeds the ordinate at X_i), the sum is called an upper sum; similarly, if, after the x_i’s are chosen, the X_i’s are chosen so that f(X_i), for each i, is a minimum in the interval [x_{i - 1}, x_i], the sum is called a lower sum. It can be proved that the upper and lower sums will have limits,  and , respectively, as the norm approaches 0. If  and  are equal and have the common value S, S is called the definite integral of f(x) from a to b and is written

The symbol ∫ is an elongated S (for sum); the f(x) dx is suggested by a term f(X_i)h_i = f(X_i) Δx_i of the sum which is used in defining the definite integral.

If y = g(x), then by differentiation y’ = g’(x). Let g’(x) = f(x), and C be any constant. Then f(x) is also the derivative of g(x) + C. The expression g(x) + C is called the antiderivative of f(x), or the indefinite integral of f(x), and it is represented by

The dual use of the term integral is justified by one of the fundamental theorems of calculus, namely, if g(x) is an antiderivative of f(x), then, under suitable restrictions on f(x) and g(x),

The process of finding either an indefinite or a definite integral of a function f(x) is called integration; the fundamental theorem relates differentiation and integration.

If the antiderivative, g(x), of f(x) is not readily obtainable or is not known, the definite integral can be approximated by the trapezoidal rule, (b - a) [f(a) + f(b)]/2 or by the more accurate Simpson’s rule:

Integral calculus involves the inverse process of finding the derivative of a function, that is, it is the process of finding the function itself when its derivative is known. For example, integral calculus makes it possible to find the equation of a curve if the slope of the tangent is known at an arbitrary point; to find distance in terms of time if the velocity (or acceleration) is known; and to find the equation of a curve if its curvature is known. Integral calculus can also be used to find the lengths of curves, the areas of plane and curved surfaces, volumes of solids of revolution, centroids, moments of inertia, and total mass and total force.