lecture 2

6. Integration

Calculus with Business Applications II Math 1690

Spring 2011

Calculus with Business Applications II Math 1690 6. Integration

6.7 Applications of the Definite Integral to Business andEconomics

Consumers’ Surplus

The consumers’ surplus is given by

CS =

∫ x̄

0D(x)dx − p̄x̄

where D is the demand function, p̄ is the unit market price, and x̄is the quantity sold.

Producers’ Surplus

The producers’ surplus is given by

PS = p̄x̄ −∫ x̄

0S(x)dx


6.7 Applications of the Definite Integral to Business andEconomics(contd.)

where S(x) is the supply function, p̄ is the unit market price, and x̄is the quantity supplied.

Example 1. The quantity demanded x (in units of a hundred) ofthe Mikado miniature cameras/week is related to the unit price p(in dollars) by

p = −0.2x2 + 80

and the quantity x (in units of a hundred) that the supplier iswilling to make available in the market is related to the unit pricep (in dollars) by

p = 0.1x2 + x + 40

If the market price is set at the equilibrium price, find theconsumers’ surplus and the producers’ surplus.



SolutionRecall that the equilibrium price is the unit price of the commoditywhen market equilibrium occurs. We determine the equilibriumprice by solving for the point of intersection of the demand curveand and supply curve. To solve the system of equations

p = −0.2x2 + 80

p = 0.1x2 + x + 40

we simply substitute the first equation into the second, obtaining

0.1x2 + x + 40 = −0.2x2 + 80

0.3x2 + x − 40 = 0

3x2 + 10x − 400 = 0

Factoring this last equation, we obtainCalculus with Business Applications II Math 1690 6. Integration


(3x + 40)(x − 10) = 0

Thus, x = −403 or x = 10. The first number lies outside the

interval of interest, so we are with the solution x = 10, with acorresponding value of

p = −0.2(10)2 + 80 = 60

Thus, the equilibrium point is (10, 60); that is,the equilibriumquantity is 6, 000, and the equilibrium price is $10. Setting themarket price at $10 per unit and using the above formula withp̄ = 60 and x̄ = 10, we find that the consumers’ surplus is given by



CS =

∫ 10

0(−0.2x2 + 80)dx − (10)(60)

=−0.2

3x3 + 80x |10

0 − 600

=−200

3+ 800 − 600

=400

3

or $40, 000.(Recall that x is measured in units of a hundred.) The producers’surplus is given by



PS = (10)(60) −∫ 10

0(0.1x2 + x + 40)dx

= 600 − (0.1

3x3 +

x2

2+ 40x)|10

0

= 600 − (100

3+ 50 + 400)

= 600 − 1450

3

=350

3

or $35,0003 .



Accumulated or Total Future Value of an Income Stream

The accumulated, or total, future value after T years of an incomestream of R(t) dollars per year, earning interest at the rate of rper year compounded continuously, is given by

A = erT

∫ T

0R(t)e−rtdt

where R(t) is Rate of income generation at any time t, r interestrate compounded continuously and T is Term.

Present Value of an Income Stream

The present value of an income stream of R(t) dollars per year,earning interest at the rate of r per year compounded continuously,is given by

PV =

∫ T

0R(t)e−rtdt



An annuity is a sequence of payments made at regular timeintervals. The time period in which these payments are made iscalled the term of the annuity.The amount an annuity is the sum of the payments plus theinterest earned.

Amount of an Annuity

The amount of an annuity is

A =mP

r(erT − 1)

where P is Size of each payment in the annuity, r interest ratecompounded continously, T term of the annuity (in years), and mnumber of payments per year.



Present Value of an Annuity

The present value of an annuity is given by

PV =mP

r(1 − e−rT )

where P, r ,T , and m are as defined earlier.

Example 2. Find the present value of an annuity if the paymentsare $80 monthly for 12 years and the account earns interest at therate of 10% per year compounded continuously.

Solution

Here P = $80, r = 0.1,T = 12,m = 12 and using the aboveformula we have the following for present value of an annuity



PV =(12)(80)

0.1(1 − e−(0.1)(12)) = 9600(1 − e−1.2)

Home Work: Section 6.7 on page 474 problems 1, 5, 6, 17.


lecture 2

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