MISC.. 2 Integration Formula for consumer surplus Income stream - revenue enters as a stream - take integral of income stream to get total revenue

MISC.

2

Integration

• Formula for consumer surplus

• Income stream

- revenue enters as a stream

- take integral of income stream to get total revenue

0

0 0

qqRdqqD

3

Integration Applications-

• Fundamental Theorem of Calculus

-

Example : applies to p.d.f.’s and c.d.f.’sRecall from Math 115a

aFbFdxxf XX

b

a X

Fundamental Theorem of Calculus. For many of the functions, f,

which occur in business applications, the derivative of with

respect to x, is f(x). This holds for any number a and any x, such that the closed interval between a and x is in the domain of f.

,)( duufx

a

4

Integration, Applications

Example 4.

The Plastic-Is-Us Toy Company

incoming revenue -as an income stream(rather than a collection of discrete payments)

At a time t years from the start of its fiscal year on July 1 the company expects to receive revenue at the rate of A(t) million dollars per year

Records from past years indicate that Plastic-Is-Us can model its revenue rate

A(t) = 110t5 + 330t4 330t3 + 110t2 +3.174 million dollars per year.

5


02468

0 0.25 0.5 0.75 1

t

A(t)

Oct. 1 Jan. 1 April 1 July 1July 1

The chief financial officer wants

to compute the total amount of revenue that Plastic-Is-Us will receive in one year.

The income stream, A(t), is a rate of change in money, given in million dollars per year.

the units along the t-axis are years

the area of a region under the graph of A(t) is given in

(millions of dollars/year)(years) = millions of dollars.

6

• Since gives the area between the t-axis and the graph of

• A(t), over the interval [0, T], it can be shown that the integral gives the total amount of money, in millions of dollars, that will be received from the income stream in the first T years.

T

dttA

0

)(

7


dollarsmillion007.5174.31103303301101

0

2345 dxxxxx

Use Integrating.xls to compute the total income received by Plastic-Is-Us during the period from 0 to 1 year. (Remember that we must use x, not t, as the variable of integration in Integrating.xls.)

8

The total revenue, in dollars, received from an income stream of A(t) dollars per year, starting now and continuing for the next T years is

given by .)(

0T

dttA

Integration, ApplicationsIntegration. Integration. Applications: page 12Applications: page 12Integration. Integration. Applications: page 12Applications: page 12

In addition to the total revenue, a company would often like to know the present value of its income stream during the next T years (0 t T), assuming that money earns interest at some annual rate r, compounded continuously.

Suppose that money earns at an annual rate, r, compounded continuously. The present dollar value of an income stream of A(t) dollars per year, starting now and continuing for the next T years is

given by .)(

0 T

tr dtetA

9


Example 5. We return to the Plastic-Is-Us Toy Company that we considered in Example 4. Recall that they have an income stream of A(t) = 110t5 + 330t4 330t3 + 110t2 +3.174 million dollars per year. The management of Plastic-Is-Us would like to know the present value of its income stream during the next year (0 t 1), assuming that money earns interest at an annual rate of 5.5%, compounded continuously.

Applying the integral formula for present value to Plastic-Is-Us, we use Integrating.xls to find that the present value of their income stream for one year, starting on July 1, is

million dollars.

879.4174.31103303301101

0

055.02345 dtetttt t

10

Integration, Calculus

the inverse connection between integration and differentiation is called the Fundamental Theorem of Calculus.




,)( duufx

a

Example 7. Let f(u) = 2 for all values of u. If x 1, then integral of f from 1 to x is the area of the region over the interval [1, x], between the u-axis and the graph of f.

11


The region whose area is represented by the integral is rectangular, with height 2 and width x 1. Hence, its area is 2(x 1) = 2x 2, and

0

1

2

3

0 1 2 3 4 5

u

f (u )

(1, 2) (x, 2)

x

2 x

duuf

1

)(

x 1

.22)(

1

xduufx

In the section Properties and Applications of Differentiation, we saw that the derivative of f(x) = mx + b is equal to m, for all values of x. Thus, the

derivative of with respect to x, is equal to 2. As predicted by the

Fundamental Theorem of Calculus, this is also the value of f(x).

The next example uses the definition of a derivative as the limit of difference quotients.

,)(

1x

duuf

12


Example 8. Recall the income stream of A(t) = 110t5 + 330t4 330t3 + 110t2 +3.174 million dollars per year that was expected by the Plastic-Is-Us toy company in Example 4 of Applications. Let G(T) be the total income that is expected during the first T years, for 0 T 1. Picking a time T = 0.5 years, we will check that the instantaneous rate of change of G(T), with respect to T, is the same as A(T).

Note that We now wish to compute G(0.5). Recall

that G(T) is approximated by the difference quotient

for small values of h. We will let h = 0.0001, and use Integrating.xls to evaluate G(0.5 + 0.0001) and G(0.5 0.0001). Integrating.xls rounds the numerical values of integrals to four decimal places. For the present calculation, we gain extra precision by copying the values from Cell N20 and keeping all of their decimal places.

G(0.5 + 0.0001) = G(0.5001) = 2.79078611562868

G(0.5 0.0001) = G(0.4999) = 2.78946381564699

.)()(

0T

dttATG

,2

)()(

h

hTGhTG

13


These give a value of 6.6115 for the difference quotient

rounded to four decimal places. This is the

instantaneous rate of change in total income after 0.5 years. Integrating.xls

shows the same value for A(0.5).

Noting that we have

verified the Fundamental Theorem of Calculus. At T = 0.5, the derivative of

with respect to T, is equal to A(T).

,)(

0T

dttA

,)()(

0T

dttATG

,0002.0

)4999.0()5001.0( GG

Normal, CalculusNormal Distributions. Normal Distributions. CalculusCalculusNormal Distributions. Normal Distributions. CalculusCalculus

4. Calculus*

The Fundamental Theorem of Calculus, that gives a connection between the two main components of calculus, differentiation and integration,

Let X be an exponential random variable with parameter = 2.

use Differentiating.xls to plot both FX(x) and its derivative for positive values of x. We also plot fX(x) for positive values of x.

Normal, Calculus

It appears that, for positive values of x, the graphs of the p.d.f., fX, and the derivative, FX, of the c.d.f. are identical.

F X (x)

0.0

0.2

0.4

0.6

0.8

1.0

0 3 6 9 12 15

x

F X (x )DERIVATIVE OF F X (x)

0.00.10.20.30.40.50.6

0 3 6 9 12 15

x

F X ' (x )

Normal Distributions.Normal Distributions.Calculus: page 2Calculus: page 2

Normal Distributions.Normal Distributions.Calculus: page 2Calculus: page 2

f X (x)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 3 6 9 12 15

x

f X (x )

Normal, CalculusNormal Distributions. Normal Distributions. Calculus: page 3Calculus: page 3Normal Distributions. Normal Distributions. Calculus: page 3Calculus: page 3

In summary, where the cumulative distribution function, FX, is differentiable, its derivative is the probability density function, fX.

Hence, the c.d.f., FX, for the continuous exponential random variable, X, is the integral of the p.d.f., fX.

.)()0(

0x

X duufxXP

duufxFx

XX )()(

0

Normal, Calculus

These relationships are not peculiar to exponential random variables. Let X be any continuous random variable.

The integral of the p.d.f., fX, is the c.d.f., FX.

Where FX is differentiable, its derivative is fX.

These can be combined to show that the derivative of

with respect to x, is fX(x).

Example:If X is a uniform random variable on the interval [0,20]. What is the derivative of

Normal Distributions. Normal Distributions. Calculus: page 4Calculus: page 4Normal Distributions. Normal Distributions. Calculus: page 4Calculus: page 4

,)(

0

duufx

X

duufx

X )(0

duufxFx

XX )()(

0

Normal, Calculus

These can be combined to show that the derivative of

with respect to x, is fX(x).

Example:If X is a uniform random variable on the interval [0,20]. What is the derivative of

•We know for uniform the p.d.f is a horizontal line between 0 and 20. here u=20, the Final Answer


,)(

0

duufx

X

20/1)( xf X

Normal, Calculus

(material continues)





,)( duufx

a

Combining this with our earlier information that

we again see that the derivative of with respect to z, is fZ(z).

This inverse relationship between integration and differentiation for probability functions is another instance of the Fundamental Theorem of Calculus, as stated previously in the section Calculus of Integration from Project 1.

,)()( duufzFz

ZZ

,)( duufz

Z

C IT

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MISC.. 2 Integration Formula for consumer surplus Income stream - revenue enters as a stream - take integral of income stream to get total revenue