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Chapter 8 – Further Applications of Integration

8.4 Applications to Economics and Biology

8.4 Applications to Economics and Biology Erickson

8.4 Applications to Economics and Biology2

Applications to Physics and Engineering Among the many applications of integral calculus to

economics and biology, we will consider these today: Consumer Surplus Blood Flow Cardiac Output

Remember, the marginal cost function C′(x) was defined to be the derivative of the cost function. Refer to sections 3.7 and 4.7.

Erickson

8.4 Applications to Economics and Biology3

Consumer Surplus The graph of a typical demand function is called a demand

curve. If X is the amount of the commodity that is available, then P = p(X) is the current selling price.

Divide the interval [0, X] into n subintervals of length x. The amount saved is

(savings per unit)(number of units) = [p(xi) – P] x

Erickson

The total savings is given by the Riemann Sum:

1

n

ii

p x P x

8.4 Applications to Economics and Biology4

Consumer Surplus If we let n ∞, the Riemann sum approaches the integral

which is called the consumer surplus for the commodity. The consumer surplus represents the amount of money saved

by consumers in purchasing the commodity at price P which corresponds to an amount demanded of X.

( )X

o

p x P dx

Erickson

8.4 Applications to Economics and Biology5

Example 1 – pg. 566 #4 The demand function for a certain commodity is

Find the consumer surplus when the sales level is 300. Illustrate by drawing the demand curve and identifying the consumer surplus as an area.

20 0.05p x

Erickson

8.4 Applications to Economics and Biology6

Blood Flow The equation is called Poiseuille’s Law, and it shows that

the flux is proportional to the 4th power of the radius of the blood vessel.

Note: F is called the flux, R is the radius of the blood vessel, l is the length of the blood vessel, P is the pressure between the ends of the blood vessel, and is the viscosity of the blood.

4

8

PRF

l

Erickson

8.4 Applications to Economics and Biology7

Cardiac Output The cardiac output of the heart is the volume of blood

pumped by the heart per unit of time, that is, the rate of flow into the aorta. The cardiac output is given by

0

( )T

AF

c t dt

Erickson

Note: F is the flow rate, A is the amount of dye known, and c(t) is the concentration of the dye at time t.

8.4 Applications to Economics and Biology8

Example 2 – pg. 567 #18 After an 8-mg injection of dye, the reading of the dye

concentration in mg/L at two second intervals are shown in the table. Use Simpson’s Rule to estimate the cardiac output.

t c(t) t c(t)

0 0 12 3.9

2 2.4 14 2.3

4 5.1 16 1.6

6 7.8 18 0.7

8 7.6 20 0

10 5.4

Erickson

8.4 Applications to Economics and Biology9

Book Resources

Erickson

Video Examples Example 1 – pg. 564 Example 2 – pg. 566

More Videos Using Integrals to find Consumer Surplus

Wolfram Demonstrations Consumer and Producer Surplus