1. By:Anmol Pasa ShresthaChhitiz ShresthaLeeza ShresthaNiraj TaujaleRaj ShresthaTenzing TashiZhang Peng

2. Introduction The relationship involving the rateof change of two variables, but also needed to know the directrelationship between the twovariables For example, we may know thevelocity of an object at a particulartime, but we may want to know theposition of the object at that time. To find this direct relationship, weneed to use the process which isopposite to differentiation. This iscalledintegration (oranti-differentiation). The processes of integration areused in many applications 3. Background Animportant concept inmathematics, Defined informally to be the netsigned area of the region in the xy-plane bounded by the graph of, the x-axis, and the vertical linesx = a and x = b May also refer to the notion ofantiderivative, a function F whosederivative is the given function the basic tools of calculus, withnumerousapplicationsinscience, business and engineering 4. 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 approximationpoints 0, 15, 25, and so on to 1. Thus 15, 25, and so on to 1 = 1. 5. 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 6. History Integration can be traced as far backas ancient Egypt before 1800 BC Further developed and employed byArchimedes and used to calculateareas for parabolas Similar methods were independentlydeveloped in China around the 3rdcentury AD by Liu Hui Next major step in integral calculuscame in Iraq when the 11th centurymathematician Ibnal-Haytham(known as Alhazen in Europe Also formulated independently byIsaac Newton and Gottfried Leibnizin the late 17th century 7. History Acquired a firmer footingwith the development oflimits and was given asuitable foundationbyCauchy in the first half ofthe 19th century, Integrationwas firstrigorously formalized, usinglimits, by Riemann , Other definitions ofintegral,extendingRiemanns and Lebesguesapproaches, wereproposed. 8. Techniques of integration Various techniques ofintegration Integration by generalrule Integration by exponentialform Integration by parts Integration by substitution Etc. 9. Methods of integration The General Power Formula: Example: 10. Methods of integration The Basic Logarithmic Form: Example 11. Methods of integration The Exponential Form: Example 12. Methods of integration Integration by Parts: Example u = ln xdv = dxv = x 13. Methods of integration Integration by Substitution: Example 14. Application of Integration The Petronas Towers in Kuala Lumpurexperience high forces due to winds.Integration was used to design thebuilding for strength. The Sydney Opera House is a veryunusual design based on slices out of aball. Many differential equations (onetype of integration) were solved in thedesign of this building. Historically,one of the first uses ofintegration was in finding the volumes ofwine-casks (which have a curvedsurface). 15. Benefits of integration inBusiness Introduce new applicationsand technologies moreefficiently and at a lower cost More easily modify andautomate business processesto meet new needs Providemore deliverychannels for your organization Replacing batch processingwith real-time communication Linking back-office systems tonew applications . Sharing data between System 16. Benefits of integration inBusiness Commercial organizations use mathematicsin accounting, Inventory management, Marketing, Sales forecasting, and Financial analysis 17. PU Board Questions (2008) 18. PU Board Questions (2009) 19. Conclusion Very important mathematical tool Used in many fields Important in business Helps to estimate things like Marginal cost Marginal revenue Profit Gross loss Etc.