By:•Anmol Pasa Shrestha•Chhitiz Shrestha•Leeza Shrestha•Niraj Taujale•Raj Shrestha•Tenzing Tashi•Zhang Peng
Introduction The relationship involving the rate of change of two variables, but also needed to know the direct relationship between the two variables For example, we may know the velocity of an object at a particular time, but we may want to know the position of the object at that time. To find this direct relationship, we need to use the process which is opposite to differentiation. This is called integration (or anti- differentiation). The processes of integration are used in many applications
Background An important concept in mathematics, Defined informally to be the net signed area of the region in the xy- plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b May also refer to the notion of antiderivative, a function F whose derivative is the given function ƒ the basic tools of calculus, with numerous applications in science, business and engineering
Background Example x = 0 to x = 1 and y = f(0) = 0 and y = f(1) = 1. Its area is exactly 1. Decreasing the width of the approximation rectangles shall give a better result; so cross the interval in five steps, using the approximation points 0, 1⁄5, 2⁄5, and so on to 1. Thus √1⁄5, √2⁄5, and so on to √1 = 1.
Background Basic knowledge of derivatives is a must Definition: The differential of y = f(x) is written: dy = f (x)dx. Example: Find the differential of y = 3x5- x. Answer dy = f(x)dx dy = (15x4 - 1)dx
History Integration can be traced as far back as ancient Egypt before 1800 BC Further developed and employed by Archimedes and used to calculate areas for parabolas Similar methods were independently developed in China around the 3rd century AD by Liu Hui Next major step in integral calculus came in Iraq when the 11th century mathematician Ibn al-Haytham (known as Alhazen in Europe Also formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century
History Acquired a firmer footing with the development of limits and was given a suitable foundation by Cauchy in the first half of the 19th century, Integration was first rigorously formalized, using limits, by Riemann , Other definitions of integral, extending Riemanns and Lebesgues approaches, were proposed.
Techniques of integration Various techniques of integration Integration by general rule Integration by exponential form Integration by parts Integration by substitution Etc.
Methods of integration The General Power Formula: Example:
Methods of integration The Basic Logarithmic Form: Example
Methods of integration The Exponential Form: Example
Methods of integration Integration by Parts: Example u = ln xdv = dxv = x
Methods of integration Integration by Substitution: Example
Application of Integration The Petronas Towers in Kuala Lumpur experience high forces due to winds. Integration was used to design the building for strength. The Sydney Opera House is a very unusual design based on slices out of a ball. Many differential equations (one type of integration) were solved in the design of this building. Historically,one of the first uses of integration was in finding the volumes of wine-casks (which have a curved surface).
Benefits of integration inBusiness Introduce new applications and technologies more efficiently and at a lower cost More easily modify and automate business processes to meet new needs Provide more delivery channels for your organization Replacing batch processing with real-time communication Linking back-office systems to new applications . Sharing data between System
Benefits of integration inBusiness Commercial organizations use mathematics in accounting, Inventory management, Marketing, Sales forecasting, and Financial analysis