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

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

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

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

5. Ekstrom Math 115b 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)

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

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

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

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

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

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

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

13. Ekstrom Math 115b Integration Midpoint Sums.xlsm

14. Ekstrom Math 115b Integration Midpoint Sums.xlsm

15. Ekstrom Math 115b Integration Midpoint Sums.xlsm

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

17. Ekstrom Math 115b Integration Integrating.xlsm

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

19. Ekstrom Math 115b 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

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

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

22. Ekstrom Math 115b 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.

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

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

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

26. Ekstrom Math 115b 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

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

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

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

30. Ekstrom Math 115b 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