apcalcApplications+of+Integrals

=toc= =**Applications of Integrals** =

Introduction[[image:area_under_curve_4.gif width="200" height="192" align="right" caption="Area bounded by two curves"]]:
An [|integral] is the total amount of area of the region in the //xy//-plane that is bounded by the graph of //ƒ//, the //x//-axis, and the vertical lines //x// = //a// and //x// = //b,// or the amount of area between two or more curves (two or more equations). An integral can also be defined as an, which refers to a function //F// whose derivative is the given function ƒ , this would usually result in an [|indefinite integral]. There is also another type of intergal: [|definite integrals]. [|Applications of integrals]is how integrals are applied to real-life situations; such as [|physical sciences], [|computer science], [|engineering], and business. Each of those types of integrals can be applied to real-life situations in different ways. For example, one of the most useful real-life applications for calculus, is for the use of physics when determining [|displacement], [|velocity] and [|acceleration]. Working from displacement to velocity, or velocity to acceleration, one would have to use the process of differentiation; however, to work from acceleration to velocity, or velocity to displacement (in other words, working backwards) one would have to use the opposite process of differentiation: [|intergration].

History:
Integration as been around as long as 1800 B.C, in Ancient Egypt, and since then it has changed over centuries. For example, in Ancient Egypt, the method of integration was "[|Moscow Mathematical Papyrus]," which had been used for determining the volume of a pyramid. In 370 B.C, [|Eudoxus] used the [|"method of exhaustion]" which found the area and volume by breaking the object up into an infinite amount of shapes that the area or volume was already known. That method was further elaborated upon by [|Archimedes], who then went on to approximate the area of circles and calculated the area of parabolas. As time went on, in the 5th century, [|Zu Chongzhi]and [|Zu Geng]used that same method to find the volume of a sphere. Later on in the same century, [|Aryabhata], an Indian mathematician, used a similar method to find the area of a cube. Several centuries later, in the 11th century, [|Ibn al-Haytham], an Iraqi mathematician, developed the "[|Alhazen's problem]." When solving this problem, he used integration to find the volume of a paraboloid. Using this method, he was able to determine the integrals up to the fourth degree. The most significant advance in integration was in the 17th century [|Newton] and [|Leibniz] made the discovery of the "[|fundamental theorem of calculus]." This theorem makes a connection between differiention and integration. This would help calculate integrals, and also allows one to solve broader problems. Last but not least, the theorem gave way for precise analysis of function with continuous domains.

Applications:
﻿Indefinite integrals can be used to find displacement from velocity and velocity from acceleration. Integrals can also be used to find the average value of a function, the area between two curves, and volumes. The average value of a function, is found by the equation: ﻿. There is also the [|mean value theorem]:, which is related to the average value. To find the area between the curves of functions, one would use the equation:. The equations used to find the volume of the curves are, with the area represented as. There are many different ways to find an integral, or an antiderivative. One can use [|riemann sums], or simply reverse the differientiation process.

Examples:
Here are a few examples on how to find the average value of a function: 



Here is an example of finding the area between the curves:





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<span style="font-family: 'Comic Sans MS',cursive;">﻿Advantages:
<span style="font-family: 'Comic Sans MS',cursive;">Some advatages are, that some ways to find integrals are more simplistic than others. For example, in some cases, te Riemann sums may be easier to use than actually finding the antiderivative itself. Other advantages are that some methods can provide exact values. This will be more accurate and be helpful when needing to determine information. Another advantage, is depending on which method one decides on, computing the information can be easier, or at least less tedious. For example, using the Riemann sums method, the person decides how many intervals they can compute. The less they chose, the easier it is.

<span style="color: #00ff00; font-family: 'Comic Sans MS',cursive;">﻿Theoretical and Applicable Issues:
<span style="background-color: transparent; color: #00ff00; font-family: 'Comic Sans MS',cursive; font-size: 16px; text-decoration: none; vertical-align: baseline;">One of the disadvanatages, is that there are many different ways that one can go about determining the integral. One way, is by using the Riemann sums, or [|Simpson's Rule]another is by using the [|trapezoidal rule], there's the integral equations itself, and other scenarios. One would need to determine which is the best method, not only for their problem at hand, but themselves as well. Another problem, is that not only of these methods are accurate. The Riemann Sums and Trapezoidal rule are approximations of integrals, in most cases, they will not provide the most accurate answer. For example, looking at the graph to the right, that graph is utilizing the Riemann Sums integration. Some of the rectangles are below the graph and others are above. In an ideal world, they would cancel each other out, making the answer correct. However, that is not always the case, and adding these rectangles will not always provide the correct answer. Even if one were to use integration, to find the antiderivative, the answer would not be 100% equal.