Integral Calculus Essay, Research Paper
Calculus is a powerful field of mathematics. Through the years, Calculus has been used to figure out extremely complicated and time-consuming problems in the fields of Physics and Engineering. The field of Calculus is divided into two major groups. The two groups are Differential Calculus and Integral Calculus. In Differential Calculus, the function is presented and the derivative or rate of change must be computed, while in Integral Calculus the derivative is given and the function must be computed. To find the function, one must go through the process of integration.The need for integration was first became prevalent during the time of Archimedes. The brilliant mathematician devised a way to calculate the area of uncommon or nonlinear figures. The only problem was that his methods were not accurate enough for some scientists. Archimedes used a method that was called the “Method of exhaustion”. In order to find the area, many triangles were inscribed into the figure until it was just about filled up or “exhausted”(Anton 297). The areas of each triangle was found and added up to give the area of the major figure. This method was a breakthrough for its time but it was not until Sir Isaac Newton and Gottfried von Leibniz that this method was perfected. They stated that if a quantity can be computed by exhaustion, then it can also be computed much more easily using the antiderivative. Through this discovery, they developed the Fundamental Theorem of Calculus (Simmons 167). In order to understand the concept of antiderivatives and integrals, one must be familiarized with the derivative.The basic definition of the derivative is the rate of change of a function (Simmons 46). Rate of change can be used to figure out many different things. One of the most common uses for the derivatives is to find the sensitivity to changes. For example, using derivatives one can see how much change will occur if the variable changes slightly. To find the rate of change, one must first know the function. A function is a process being done or usually it is described by an equation. In an equation the function is usually stated as being f(x) or read as f at x (Kleppner and Ramsey 64). Using the function as your dependent variable you can simplify it by calling it y. The main mathematical equation used for find the derivative of a function is: Y = D Y limDx®0 DX The term limit (lim.) is very important to Calculus. In the equation the limit was used to get the points between X1 and X2 as small as possible. So we want the DX (X2 – X1) to approach zero. This would give us a much more accurate reading. If DX was not limited to zero than we would get a lot of unwanted and vague information (Leithold 172). The other way to describe the mathematical equation of the derivative is: Y= F(x +Dx) f(x) lim.Dx®0 DX f(x + Dx) is the final result of the y-function with the change and f(x) is the original or initial function. Subtracting f(x) from f(x + Dx) gives you DY (Simmons 46). After understanding this concept, one can now learn to integrate. As stated earlier, integration is the process where one knows the derivative and must find the original function. Basically, it is the accumulation of all the changes. To better understand this concept, consider a continuous curve y = f(x) lying above the x-axis and let (a , b) be an interval on the x-axis. We can then denote the function to be A(x). It has already been proven that A (x) = f(x). This proves that finding the area function reduces to doing the opposite of the differentiation process and recovering A(x) from its known derivative f(x). Once A(x) is found, the area under the curve y = f(x) over a specific interval can be found by evaluating the specific y values (Anton 297). Y – axis Y = f(x) A(x) A B X – Axis Another proven fact about derivatives is that the derivative of a constant is zero. This causes a problem because one cannot find the exact function because it is not known if the original function ever contained a constant. If we suspect a constant was originally in the function we can then denote it writing + C at the end This usually occurs when the limit is between -¥ ® ¥. In other words, this interval contains every number. Functions that contain constants are called indefinite integrals. The definition explains the term very nicely because if we did not know whether the initial function had a constant or not, we would be indefinite in our conclusions (Anton 299). When dealing with integrals there are a few principals that must be used. For example, if d [F(x)] = f(x) dx then the functions of the form F(x) + C are antiderivatives of f(x). This can be denoted by writing: ò f(x) dx = F(x) + CThe symbol ò is called an integral sign. So the integral of a function is written ò f(x) and it would be equal to F(x) + C. The capital F means that it is an antiderivative. Since this function contains a constant than it is considered an indefinite function and the constant is known as the constant of integration. Also, the symbol dx serves to identify the independent variable. The main difficulty in understanding integrals is that it requires a lot of guesswork. By looking only at the derivative of a function we try to guess the function itself, because there is no direct definition to an integral. In order to simplify the guesswork process, we need to keep in mind that every differentiation formula produces a companion integration formula. In order to assist in the guesswork, many other properties are utilized used (Anton 302). For example, a constant factor can be moved through an integral sign:
ò nf(x) dx = n ò f(x) dx. Also, an antiderivative of a sum is the sum of the antiderivatives: ò [ f(x) + g(x)] dx = ò f(x) dx + ò g(x) dx. To help with the integration process many tables have been created to show some basic integration formulas. The following is a table that contains basic and common integration formulas. Differentiation formula Integration Formula 1. d [x] = 1 ò 1 dx = x + C dx 2. d x r+1 = x r ( r ¹ -1) ò x r dx = x r + 1 + C ( r ¹ -1) dx r +1 r + 1 In order to evaluate the definite integral of a continuous function, Newton and Leibniz devised an approach that would seem to make the problem a lot more difficult than it really was. This theorem would eventually develop into the Fundamental Theorem of Calculus. The theorem stated that in order to solve receive fixed area answer, we must first use a variable area produced when the right side of the border is considered to be moveable. So it seems that in order to solve this problem, we replace it by an apparently more difficult problem. The theorem’s intent was to get the area of the continuous function and then subtract it from the portion to the right of the initial function. This way we would be subtracting one indefinite integral from another to receive a definite integral answer (Simmons 167). The Fundamental Theorem of Calculus formally states that if f(x) is a continuous function on a closed interval [a, b], and if F(x) is any antiderivative of f(x), so that (d/dx) F(x) = f(x). This could also be expressed as: When dealing with the areas under a curve, many times the problem of adding number after number comes up. These numbers usually accumulate time after time again. This can be a time consuming process. In order to simplify this process the term sigma notation å is used (Anton 315). Sigma notation is also sometimes referred to as summation notation. For example if f(x) is a function of n, and a and b are integers such that a £ b, then: b å f(n) n=a To show how useful this notation is take for example the sum: 15 + 25 + 35 + 45 . In sigma notation this can be simply written as 4 å n5 n = 1 Sigma notation has become essential to mathematicians, businessmen, and physicists whose numbers sometimes are added together hundreds of times. This form of notation accumulates and organizes this set of sums very efficiently. It is also possible to determine the area under a curve using limits. This concept is very similar to Archimedes discovery only it adds the concept of limits to it. In this concept many rectangles are inscribed under the curve. Each of these rectangles are congruent in width. Rectangles were chosen because the area equation is already known and has been proven. The only problem with rectangles is that because of their 90 degrees edges, they can not fill in all the gaps underneath a curve. Using limits however, we can limit the width of the rectangles to zero. The width can not literally be zero, but they become closest thing to it. When the width of these rectangles are very close to zero, only a small portion of the area under a curve is being left out. With limits, this portion is almost insignificant. After the rectangles are limited to zero, we can now utilize å notation to simply accumulate all the rectangles (Simmons 162). It can be stated that the approximated sum (s n) of all these rectangles is: n Sn = lim å f(x) Dxk . n®¥ k =1 Calculus is a major tool in all sciences, especially physics and engineering. Much of the original inspiration for the development of calculus came from the science of mechanics. Mechanics rests on certain basic principles that were first introduced by Newton. All of these principles require the concept of the derivative and its counterpart, integration (Boyer 187).Here are a few example of the importance of Calculus. Calculus is truly a powerful field of mathematics that has helped the human race technologically. It has brought man out in space as well as deep in the ocean. Most fields of math are the works of many great geniuses. In Calculus, it is truly the work of two of the greatest minds of all time. Sir Isaac Newton and Gottfried Wilhelm von Leibniz are credited with being the inventors or discoverers of Calculus. Newton and Leibniz were able to take all kinds of natural occurrences and formulate them into mathematical equations. Even though they were living during the same time, Newton and Leibniz did not work together, the Calculus that we know today is a combination of their works (Boyer 189). These two great minds were able to see into the future by creating all kinds of mathematical equations for the good of mankind. Who knows what lies next?