Class 9: Class 9: Area, Consumer Surplus, IntegrationArea, Consumer Surplus, Integration

-1.2

-10 q

D(q)

Demand Function

Revenue

q

D(q)

What is Total Possible Revenue?What is Total Possible Revenue?

-1.2

-8

Demand Function

Total PossibleRevenue

The total possible revenue is the money that the producer would receive if everyone who wanted the good, bought it at the maximum price that he or she was willing to pay.

Consumer SurplusConsumer Surplus

-1.2

-8q

D(q)

Revenue

ConsumerSurplus

NotSold

Demand Function

The total extra amount of money that people who bought the good would have been willing to pay is called the consumer surplus.

Finding AreasFinding Areas

What is the area of the region R that is enclosed between the x-axis and the graph of f(x) = 2x x2/2, for x between 1 and 4?

1 4

2

1 R

Finding AreasFinding Areas

What is the area of the region R that is enclosed between the x-axis and the graph of f(x) = 2x x2/2, for x between 1 and 4?

1 4

2

1 R

For n = 6 rectangles

1 4

2

1

1.5 2 2.5 3 3.5

1 4

2

1

x0 x6x2x1 x3 x4 x5

m1 m6m2 m3 m4 m5

f m1( )

{x

Sum of areas called S6

= f(m1)x + f(m2)x + f(m3)x + f(m4)x + f(m5)x + f(m6)x

= 4.531250.

More rectangles: Larger n

1 4

2

1xmfS

n

iin

1

)(

As n increases, the value of Sn increases, getting closer and closer to the true are under the curve.

Integral NotationIntegral Notation

We write the value of the midpoint sum as n gets very large by an integral

bx ax xfdxxfb

a and between )(under Area)(

Find Revenue from Buffalo Dinners Not Sold:Optimum price and quantity are $19.19 and 2300

0 1000 2000 3000 4000 5000

$8

$16

$24

$32

NotSold

ConsumerSurplus

q = 2,300

D(2,300) = $19.99

Revenue$45,977

2,300

$19.99 Demand Function

D(q) = 0.0000018q2 0.0002953q + 30.19

$0

014,4

300,2

643,18$)(SoldNot dqqD

4014

Find Consumer Surplus for Dinners

0 1000 2000 3000 4000 5000

$8

$16

$24

$32

NotSold

ConsumerSurplus

q = 2,300

D(2,300) = $19.99

Revenue$45,977

2,300

$19.99 Demand Function

D(q) = 0.0000018q2 0.0002953q + 30.19

$0

.379,15$977,45$356,61$)()(Surplus

Consumer300,2

0

qRdqqD