application of integration to economy and biology exercise

application of integration to economy and biology exercise

1. PROJECT Topic : Application of Integration to Economics and Biology,Exercises Subject : Mathematic

2. The application of integration to economics is based is four concepts: Marginal and total revenue, cost, and profit; Capital accumulation over a specified period of time; Consumer and producer surplus; Lorenz curve and Gini coefficient; On the other hand the application of integration to biology is based in two concepts : Blood flow Cardiac Output

3. Consumer Surplus Economic surplus refers to two related quantities. Consumer surplus is the monetary gain obtained by consumers, they are able to buy something for less than they had planned on spending. In calculus terms, consumer surplus is the derivative of the definite integral of the demand function with respect to price, from the market price to the maximum reservation price.

4. Supply and Demand Chart: Graph illustrating consumer (red) and producer (blue) surpluses on a supply and demand chart. CS = where D(P) is a demand curve as a function of price.

5. We can focus on the producer side . The area under the supply function, from 0 to the quantity sold, measures the producers’ need for revenue. The area in the rectangle with that same base and height equal to the sale price , measures the actual producer revenue. The difference between the two is a quantity we will call  producer surplus . In Calculus terms : Needed revenue: Producer revenue =   Producer surplus =

6. Lorenz Curves and the Gini Index (Part 1) Lorenz curve is defined by a function L(x),with 0 ≤ x ≤ 1 that measures the proportion of something that is held by the bottom x proportion of the population. Since, a person cannot have negative income, The Lorenz functions are nonnegative and increasing and since the functions are measured from the bottom, we also have L(x) ≤ x for all x. We can make a few more observations. The population as a whole has the entire income of the population. An empty set of the population has none of the population's income. Any bottom segment will have nonnegative income. In formulas these observations become  L(1)=1, L(0)=0 and L(x) ≥ 0, for all x.

7. Lorenz Curves and the Gini Index (Part 2) If we had perfect equity, our Lorenz function would be L(x)=x. Any Lorenz curve we find for a real population will be below this curve. The Gini index (or Gini coefficient) measures the percentage that a real Lorenz curve is below the ideal curve. Computationally : In practice, the Gini index is an application where a numeric approximation of an integral is the method most likely to be used.There is no good model for how the income will be distributed, we can simply connect the points with line segments and find the area using the area formula for a trapezoid.

8. Example 2 Consider the Lorenz curve L(x) = 0.3x+0.7 x 4 . Find the Gini index. Solution: We need to use the formula that we mentioned below to find the Gini index . G= G=

9. There is a cylindrical symmetry in the blood vessel, we first consider the volume of blood passing through a ring with inner radius r and outer radius( r+dr ) per unit time ( dF ) : dF = (2 where v(r) is the speed of blood at radius r. Here, 2 dr is the area of the ring. Therefore, the total flux F is written as: F= where R  the radius of the blood vessel. Once we have an (approximate) expression for v(r),we can calculate the flux from the integral. Blood flow

10. Blood Flow: (a) A tube; (b) The blood flow close to the edge of the tube is slower than that near the center.

11. Cardiac Output The cardiac output of the heart is the volume of blood pumped by the heart per unit time, that is, the rate of flow into the aorta. Let be the concentration of the dye at time. If we divide into subintervals of equal length,then the amount of dye that flows past the measuring point during the subinterval from to is approximately : (concentration)(volume) = c(t 1 )(F Δt )

12. Where F is the rate of flow that we are trying to determine. Thus the total amount of dye is approximately : and, letting n ∞ we find that the amount of dye is : Thus the cardiac output is given by : where the amount of dye is known and the integral can be approximated from the concentration readings.

13. Exercise 10 page 591 A movie theater has been charging $10.00 per person and selling about 500 tickets on a typical weeknight. After surveying their customers, the theater management estimates that for every 50 cents that they lower the price, the number of movie goers will increase by 50 per night. Find the demand function and calculate the consumer surplus when the tickets are priced at $8.00.

14. Solution: p(500)=$10.00. The lowering price of tickets is as follows : 50 cents*($1/100 cents)=$0.5 When tickets increase by 50 => p(550)=$(10-0.5)=$ 9.50 Now we can find the demand function based on the slope equation y-y 1 =m(x-x 1 )

15. y-10= (x-500) => => y=-0.01x+15=p(x) Now we can find the number of tickets sold at P=$8 8=-0.01+15 0.01x=7 x=700

16. The consumer surplus is the grey region . We can take advantage of the triangle shape (finding its area). b

17. But we can also calculate it by using standard formula of thr Area for the triangle A=b*h ∕ 2 b=700 units h=15-8=7 units We replace these values on our formula A=700*7/2=2450

18. References: https://mathstat.slu.edu/~ may/ExcelCalculus/sec-7-8-BusinessApplicationsIntegral.html Calculus Metrix Version ,James Stewart

19. Worked by : Fatjona Jangulli Kejsi Cypi Klea Canaj Mikea Toma Mirela Bendo Polikseni Dema Class: IE 101