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Integration

Mathematics for Business Decisions, part II. Integration. Math 115b. Ekstrom Math 115b. Integration . Demand Function. D ( q ).

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Integration

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  1. Mathematics for Business Decisions, part II Integration Math 115b Ekstrom Math 115b

  2. Integration Demand Function D(q) Revenue D(q) q q Motivation • Revenue as an area under Demand function • . Ekstrom Math 115b

  3. Integration Demand Function Total Possible Revenue Total Revenue • Total possible revenue is the revenue gained by charging the max price per customer Ekstrom Math 115b

  4. Integration Demand Function Consumer Surplus D(q) Revenue Not Sold q Revenue • Consumer surplus – revenue lost by charging less • Producer surplus – revenue lost by charging more (i.e. “not sold” revenue) Ekstrom Math 115b

  5. Integration Approx. area under curve • Counting rectangles (by hand) • Using midpoint sums (by hand) • Using Midpoint Sums.xlsm (using Excel) • Using Integrating.xlsm (using Excel) Ekstrom Math 115b

  6. Integration Counting Rectangles Ex. Approx. 9 rectangles Each rectangle is 0.25 square units Total area is approx. 2.25 square units Ekstrom Math 115b

  7. Integration Midpoint Sums • Notation • Meaning Ekstrom Math 115b

  8. Integration Midpoint Sums • Process Find endpoints of each subinterval Find midpoint of each subinterval Ekstrom Math 115b

  9. Integration Midpoint Sums • Process (continued) • Find function value at each midpoint • Multiply each by and add them all • This sum is equal to Ekstrom Math 115b

  10. Integration Midpoint Sums • Ex. Determine where . Ekstrom Math 115b

  11. Integration Midpoint Sums • Ex. (Continued) Ekstrom Math 115b

  12. Integration Consumer Surplus • Ex. (Continued) Ekstrom Math 115b

  13. Integration Midpoint Sums.xlsm Ekstrom Math 115b

  14. Integration Midpoint Sums.xlsm Ekstrom Math 115b

  15. Integration Midpoint Sums.xlsm Ekstrom Math 115b

  16. Integration Integrating.xlsm • File is similar to Midpoint Sums.xlsm • Notation: or or…. Ekstrom Math 115b

  17. Integration Integrating.xlsm Ekstrom Math 115b

  18. Integration Integrating.xlsm • Ex. Use Integrating.xlsm to compute Ekstrom Math 115b

  19. Integration Integrating.xlsm • Ex. (Continued) • So . Note that is the p.d.f. of an exponential random variable with parameter . This area could be calculated using the c.d.f. function Ekstrom Math 115b

  20. Integration Integrating.xlsm • Ex.(Continued) Ekstrom Math 115b

  21. Integration Signed Area • Values from Midpoint Sums.xlsm can be positive, negative, or zero. • Values from Integrating.xlsm can be positive, negative, or zero. Ekstrom Math 115b

  22. Integration Consumer Surplus • Ex. Suppose a demand function was found to be: • Determine the consumer surplus at a quantity of 400 units produced and sold. Ekstrom Math 115b

  23. Integration Consumer Surplus • Ex. (Continued) Total Revenue at 400 units produced and sold Ekstrom Math 115b

  24. Integration Consumer Surplus • Ex.(Continued) Ekstrom Math 115b

  25. Integration Consumer Surplus • Ex. (Continued) • Calculate Revenue at 400 units: Ekstrom Math 115b

  26. Integration Consumer Surplus • Ex. (Continued) • Take total revenue possible and subtract revenue at 400 units • $107,508.80 - $83,569.60 = $23,939.20 • So the consumer surplus is $23,939.20 Ekstrom Math 115b

  27. Integration Consumer Surplus • Formula for consumer surplus: Ekstrom Math 115b

  28. Integration Integration Application • Income Stream • revenue enters as a stream • take integral of income stream to get total revenue/income Ekstrom Math 115b

  29. Integration Fundamental Theorem of Calculus • The derivative of with respect to x is • applies to p.d.f.’s and c.d.f.’s Ekstrom Math 115b

  30. Integration Project (What to do) • Calculate the consumer surplus to answer Question #5 • Use Integrating.xlsm (watch units) • = 459.99 - 360.86 = $99.13 million Ekstrom Math 115b

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