Integral (mathematics)
Integral calculus is a fundamental branch of mathematics that focuses on two main problems: the antiderivative problem, which seeks to find a function when its derivative is known, and the area problem, which aims to calculate the area between a function's graph and the x-axis over a specific interval. The roots of integral calculus date back to the third century BC with Archimedes' method for calculating areas, but it was during the seventeenth century that significant advancements were made independently by Sir Isaac Newton and Gottfried Leibniz, who laid the groundwork for modern calculus.
The notation for integrals, with the integral sign ∫ and the differential dx, is primarily attributed to Leibniz and is widely used today. Indefinite integrals, or antiderivatives, are defined as a family of functions that yield the same derivative when differentiated, highlighting that they are not unique. Definite integrals, on the other hand, represent the exact area under a curve and are approached through methods involving the summation of areas of rectangles within a defined interval.
A crucial link between the two types of integrals is established by the fundamental theorem of calculus, which reveals that if a function is continuous over an interval, the antiderivative can be used to determine the definite integral over that interval. This theorem emphasizes the interconnected nature of integration, showcasing its significance in both theoretical and practical applications within mathematics and beyond.
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Integral (mathematics)
The development of integral calculus was motivated by two seemingly disparate mathematical problems: finding a function whose derivative is known; and finding the area between the graph of a function and the x-axis over an interval [a,b]. The first is called the antiderivative problem, and the second is called the area problem.
Integral calculus, or integration, has been studied since the third century BC when Archimedes presented his method of exhaustion to calculate the area of a circle. Integral calculus, as it is known today, came into sharper focus in the seventeenth century AD with the works of Sir Isaac Newton and Gottfried Leibniz, who each developed modern calculus independently.
Leibniz' notation is the most widely used one today, and that is what will be used in this article. The indefinite integral, or antiderivative, of a function f(x) is denoted by:
where ∫ is an integral sign, and dx is the differential of x. This integral arises in the first of the two problems mentioned above. For the definite integral of a function on an interval [a,b], the notation is:
This integral arises in the second of the two aforementioned problems.
Overview
From differential calculus it is known that 2x is the unique derivative of x2. Antidifferentiation can be loosely thought of as an inverse operation to differentiation. This means that x2 is an antiderivative of 2x, because
. However, x2 is not the only antiderivative of 2x, because
also, so
is also an antiderivative of 2x. In fact, for any constant C,
is an antiderivative of 2x. This implies that antiderivatives are not unique, as derivatives are. In general terms the antiderivative problem can be stated as follows: Given f(x) , find a family of functions y = F(x) + C such that
. The problem is approached as follows.
The solution is on the right, as
. The final result is:
Using (1), the earlier example can be rewritten as follows.
Now consider the problem of finding the area A between the graph of
and the x-axis on the interval [0,2]. The problem is attacked by filling the region with rectangles. See Figure 1.
The base of each rectangle is
and the heights of the rectangles are
The area A is approximated by summing the areas of the rectangles.
The exact area A is found by taking the limit as the number of rectangles
, which gives precisely the definite integral of
over [0,2]:
In general, let f(x) be a function on [a,b], and let [a,b] be divided into n subintervals by a partition
, given as follows.
Let
be any point in the
subinterval,
be the width of the
subinterval, and
be the width of the widest subinterval. Then the definite integral of f on [a,b] is defined as:
provided the limit exists.
On the surface it appears that the indefinite and definite integrals have nothing to do with each other apart from the fact that they both have the integral sign. But the fundamental theorem of calculus teaches us otherwise. It states that if f is continuous on [a,b] and if F is an antiderivative of f on [a,b], then:
The significance of this theorem is that it unites the concepts of definite and indefinite integration, and its importance cannot be overstated. As an example of an application of (2), the exact area of the region in the last paragraph is found as follows.
Bibliography
Anton, Howard, Irl Bivens, and Stephen Davis. Calculus. Hoboken, NJ: Wiley, 2012.
Archimedes, and Thomas L. Heath. The Works of Archimedes. Mineola, NY: Dover, 2002.
LaTorre, D. R. Calculus Concepts : An Informal Approach to the Mathematics of Change. Boston: Cengage, 2012.
Larson, Ron, and Bruce H. Edwards. Calculus. Boston: Cengage, 2014.
Ross, Kenneth A. Elementary Analysis: The Theory of Calculus. New York: Springer, 2013.
Stewart, James. Calculus : Early Transcendentals. Belmont, CA: Cengage, 2012.