Integral Calculus Essay Research Paper Calculus is

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 produce