1. Chapter 6 Additional Integration Topics
Section 2
Applications in Business and Economics

2. Learning Objectives for Section 6.2 Applications in Business/Economics
1. The student will be able to construct and interpret probability density functions.
2. The student will be able to evaluate a continuous income stream.
3. The student will be able to evaluate the future value of a continuous income stream.
4. The student will be able to evaluate consumers’ and producers’ surplus.

3. Probability Density Functions
A probability density function must satisfy:
1. f (x) ≥ 0 for all x
2. The area under the graph of f (x) over the interval (–∞, ∞) is 1
3. If [c, d] is a subinterval of (–∞, ∞) then Probability (c ≤ x ≤ d) =

4. Probability Density Functions (continued)
Sample probability density function

5. Example
In a certain city, the daily use of water in hundreds of gallons per household is a continuous random variable with probability density function
Find the probability that a household chosen at random will use between 300 and 600 gallons.

6. Insight
The probability that a household in the previous example uses exactly 300 gallons is given by:
In fact, for any continuous random variable x with probability density function f (x), the probability that x is exactly equal to a constant c is equal to 0.

7. Continuous Income Stream
Total Income for a Continuous Income Stream:
If f (t) is the rate of flow of a continuous income stream, the total income produced during the time period from t = a to t = b is
a Total Income b

8. Example
Find the total income produced by a continuous income stream in the first 2 years if the rate of flow is
f (t) = 600 e 0.06t

9. Example
Find the total income produced by a continuous income stream in the first 2 years if the rate of flow is
f (t) = 600 e 0.06t

10. Future Value of a Continuous Income Stream
From previous work we are familiar with the continuous compound interest formula
A = Pert.
If f (t) is the rate of flow of a continuous income stream, 0 ≤ t ≤ T, and if the income is continuously invested at a rate r compounded continuously, the the future value FV at the end of T years is given by

11. Example Let’s continue the previous example where f (t) = 600 e0.06 t
Find the future value in 2 years at a rate of 10%.

12. Example Let’s continue the previous example where f (t) = 600 e0.06 t
Find the future value in 2 years at a rate of 10%.
r = 0.10, T = 2, f (t) = 600 e 0.06t

13. Consumers’ Surplus
If is a point on the graph of the price-demand equation
P = D(x), the consumers’ surplus CS at a price level of is
which is the area between p =
and p = D(x) from x = 0 to x = CS x p
The consumers’ surplus represents the total savings to consumers who are willing to pay more than for the product but are still able to buy the product for .

14. Example
Find the consumers’ surplus at a price level of for the price-demand equation
p = D (x) = 200 – 0.02x

15. Example
Find the consumers’ surplus at a price level of for the price-demand equation
p = D (x) = 200 – 0.02x
Step 1. Find the demand when the price is

16. Example (continued)
Step 2. Find the consumers’ surplus:

17. Producers’ Surplus
If is a point on the graph of the price-supply equation p = S(x), then the producers’ surplus PS at a price level of is x p CS
which is the area between and p = S(x) from x = 0 to
The producers’ surplus represents the total gain to producers who are willing to supply units at a lower price than but are able to sell them at price .

18. Example
Find the producers’ surplus at a price level of for the price-supply equation
p = S(x) = x

19. Example
Find the producers’ surplus at a price level of for the price-supply equation
p = S(x) = x x2
Step 1. Find , the supply when the price is
Solving for using a graphing utility:

20. Example (continued)
Step 2. Find the producers’ surplus:

21. Summary We learned how to use a probability density function.
We defined and used a continuous income stream.
We found the future value of a continuous income stream.
We defined and calculated a consumer’s surplus.
We defined and calculated a producer’s surplus.