Integration

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Assignment, Business Mathematics II, BBA-BI 2nd semester, Ace Institute of Management

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Integration

  1. 1. By:•Anmol Pasa Shrestha•Chhitiz Shrestha•Leeza Shrestha•Niraj Taujale•Raj Shrestha•Tenzing Tashi•Zhang Peng
  2. 2. 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
  3. 3. 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
  4. 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 approximation points 0, 1⁄5, 2⁄5, and so on to 1.  Thus √1⁄5, √2⁄5, and so on to √1 = 1.
  5. 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. 6. 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
  7. 7. 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.
  8. 8. Techniques of integration Various techniques of integration Integration by general rule Integration by exponential form Integration by parts Integration by substitution Etc.
  9. 9. Methods of integration The General Power Formula: Example:
  10. 10. Methods of integration The Basic Logarithmic Form: Example
  11. 11. Methods of integration The Exponential Form: Example
  12. 12. Methods of integration Integration by Parts: Example  u = ln xdv = dxv = x
  13. 13. Methods of integration Integration by Substitution: Example
  14. 14. 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).
  15. 15. 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
  16. 16. Benefits of integration inBusiness Commercial organizations use mathematics in accounting, Inventory management, Marketing, Sales forecasting, and Financial analysis
  17. 17. PU Board Questions (2008)
  18. 18. PU Board Questions (2009)
  19. 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.

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