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Topic 10: Topic 10: IntegrationIntegration

Jacques Jacques

Indefinate Integration 6.1Indefinate Integration 6.1

Definate Integration 6.2Definate Integration 6.2

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IntuitionIntuition

y = F (x) = xn + c

dy/dx = F`(x) = f(x) = n xn-1 Given the derivative f(x), what is F(x) ? (Integral, Anti-derivative or the Primitive function). The process of finding F(x) is integration.

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DefinitionDefinitionJust as f(x) = derivative of F(x)

dxxfxF )()(

Example

cxdxxxF 323)(

c=constant of integration (since derivative of c=0)of course, c may be =0….., but it may not check: if y = x3 + c then dy/dx = 3x2

or if c=0, so y = x3 then dy/dx = 3x2

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Rule 1 of Integration:

cxn

dxxxF nn

1

1

1)(

cxdxxxF 32

3

1)(

check: if y = 1/3 x

3 + c then dy/dx = x2

cxdxxdx.dx)x(F 01

check: if y = x + c then dy/dx = 1

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Rule 2 of Integration:

dxxfadxxafxF )()()(

Examples

cxcx..dxxdxx)x(F

3322

3

1333

check…..

caxdxadx.a)x(F

check…

cxdxdx)x(F 444 check

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• Rule 3 of Integration:

• Example

dxxgdxxfdxxgxfxF )()()()()(

cxxdxxdxxdxxxxF 2322 2323)(

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Calculating Marginal Functions

•Given MR and MC use integration to find TR and TC

dQ

TRdMR

dQ

TCdMC

dQQMRQTR .

dQQMCQTC .

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Marginal Cost FunctionMarginal Cost Function Given the Marginal Cost Function, derive an expression for Total Cost?

MC = f (Q) = a + bQ + cQ2

dQcQbQa)Q(TC 2

dQQcdQQbdQa)Q(TC 2

FQc

Qb

aQ)Q(TC 32

32 F = the constant of integration If Q=0, then TC=F F= Fixed Cost….. (or TC when Q=0)

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Another ExampleAnother Example MC = f (Q) = Q + 5 Find an expression for Total Cost in terms of Q, if TC = 20 when production is zero.

dQQ)Q(TC 5 dQdQQ)Q(TC 5

FQQ)Q(TC 52

1 2

F = the constant of integration

If Q=0, then TC = F = Fixed Cost So if TC = 20 when Q=0, then F=20

So, 2052

1 2 QQ)Q(TC

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Another ExampleAnother Example

Given Marginal Revenue, MR = f (Q) = 20 – 2Q Find the Total Revenue function?

MR = f (Q) = 20 – 2Q

dQQ)Q(TR 220

QdQdQ)Q(TR 220

cQQ)Q(TR 220 c = the constant of integration

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Example: Given MC=2Q2 – 6Q + 6; MR = 22 – 2Q; and Fixed Cost =0. Find total profit for profit maximising firm when MR=MC?

1) Find profit max output Q where MR = MC MR=MC so 22 – 2Q = 2Q2 – 6Q + 6 gives Q2 – 2Q – 8 = 0 (Q - 4)(Q + 8) = 0 so Q = +4 or Q =-2 Q = +4

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2) Find TR and TC

dQQ)Q(TR 222

QdQdQ)Q(TR 222

cQQ)Q(TR 222 so TR = 22Q – Q2

MC = f (Q) = 2Q2 – 6Q + 6

dQQQ)Q(TC 662 2

dQQdQdQQ)Q(TC 662 2

FQQQ)Q(TC 633

2 23

F = Fixed Cost = 0 (from question)

so….QQQ)Q(TC 63

3

2 23

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3. Find profit = TR-TC, by substituting in value of q* when MR = MC Profit = TR – TC TR if q*=4: 22(4) - 42 = 88-16 = 72 TC if q* =4: 2/3 (4)3 – 3(4)2 + 6(4) = 2/3(64) – 48 + 24 = 182/3

Total profit when producing at MR=MC so q*=4 is TR – TC = 72 - 182/3 = 53 1/3

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Some general points for answering Some general points for answering these types of questionsthese types of questions

Given a MR and MC curves - can find profit maximising output q* where

MR = MC - can find TR and TC by integrating MR

and MC - substitute in value q* into TR and TC to

find a value for TR and TC. then….. - since profit = TR – TC can find (i) profit if given value for F or (ii) F if given value for profit

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T h e d e f i n i t e i n t e g r a l o f f ( x ) b e t w e e n v a l u e s a a n d b i s :

)()()()( aFbFdxxfxFb

a

ba

Definite Integration

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1 ) 3

7)1(

3

1)2(

3

1

3

1 332

1

32

1

2

xdxx

2 ) 12)2(3)6(333 6

2

6

2

xdx

Example

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DefinitionDefinition

The definite integral b

a

dx)x(f can be

interpreted as the area bounded by the graph of f(x), the x-axis, and vertical lines x=a and x=b

x

f(x)

a b

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Consumer SurplusConsumer Surplus

Q

Demand Curve: P = f(Q)

Q1

P

P1

0

a

x

Consumer Surplus

Difference between value to consumers and to the market…. Represented by the area under the Demand curve and over the Price line…..

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Or more formally….Or more formally….

C S ( Q ) = o Q 1 a x - o Q 1 a P 1 W h e r e o Q 1 a x r e p r e s e n t s t h e e n t i r e a r e a u n d e r t h e d e m a n d c u r v e u p t o Q 1 a n d o Q 1 a P 1 r e p r e s e n t s t h e a r e a i n t h e r e c t a n g l e , u n d e r t h e p r i c e l i n e u p t o Q 1 H e n c e ,

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0

)()(1

QPdQQDQCSQ

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Producer SurplusProducer Surplus

Q

Supply Curve: P = g(Q)

Q1

P

P1

0

a

y

Producer Surplus

Difference between market value and total cost to producers…. Represented by the area over the Supply curve and under the Price line…..

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Or more formally….Or more formally….

P S ( Q ) = o Q 1 a P 1 - o Q 1 a y W h e r e o Q 1 a P 1 r e p r e s e n t s t h e a r e a o f t h e e n t i r e r e c t a n g l e u n d e r t h e p r i c e l i n e u p t o Q 1 a n d o Q 1 a y r e p r e s e n t s t h e a r e a u n d e r t h e S u p p l y c u r v e u p t o Q 1 H e n c e

dQ)Q(SQP)Q(PSQ

1

011

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Example 1…..Example 1…..

Find a measure of consumer surplus at Q = 5, for the demand function P = 30 – 4Q Solution

1) solve for P at Q = 5 If Q = 5, then P = 30 – 4(5) = 10

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Q

Demand Curve: P = f(Q) = 30 – 4Q

Q1 = 5

P

P1=10

0

30

Consumer Surplus

7.5

The picture….2) ‘sketch’ diagram

P = 30 – 4Q intercepts: (0, 30) and (7.5, 0)

At Q = 5, we have P = 10 ….. Draw in price line….

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Calculation… Calculation…

3 ) E v a l u a t e C o n s u m e r S u r p l u s i ) E n t i r e a r e a u n d e r d e m a n d c u r v e b e t w e e n 0 a n d Q 1 = 5 :

1000)25(2)5(30

230)430(5

02

5

0

QQdQQ

i i ) t o t a l r e v e n u e = a r e a u n d e r p r i c e l i n e a t P 1 = 1 0 , b e t w e e n Q = 0 a n d Q 1 = 5 i s P 1 Q 1 = 5 0

i i i ) S o C S = 1 0 0 – p 1 Q 1 = 1 0 0 – ( 1 0 * 5 ) = 5 0

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0

)()(1

QPdQQDQCSQ

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Example 2Example 2

If p = 3 + Q2 is the supply curve, find a measure of producer surplus at Q = 4 Solution 1) evaluate P at Q = 4 If Q = 4, then p = 3 + 16 = 19

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The picture….The picture….

Q

Supply Curve: P = g(Q) = 3 + Q2

Q1 = 4

P

P1 = 19

0

3

Producer Surplus

2) ‘Sketch’ the diagramP = 3 + Q2 intercept: (0, 3) Price line at Q = 4, P = 19

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Calculation…Calculation…

3 ) E v a lu a te P r o d u c er s S u r p lu s i) E n tire a rea u n d e r su p p ly cu rv e b e tw e en Q = 0 an d Q 1 = 4 … ..

313

4

0

34

0

2

330)4(3

1)4(3

3

13)3(

QQdQQ

ii) to ta l rev en u e = a rea u n d e r p rice lin e (p 1 = 1 9 ), b e tw e en Q = 0 an d Q 1 = 4 , an d th is = p 1Q 1 = 7 6

iii) S o P S = p 1Q 1 – 3 3 1 /3 =

7 6 – 3 3 1 /3 = 4 2 2 /3

dQ)Q(SQP)Q(PSQ

1

011

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Example 3Example 3• The inverse demand and supply functions

for a good are, respectively:• and

• Find the market equilibrium values of P and Q.

• Find the Total surplus (CS + PS) when the market is in equilibrium.

142 QP 2QP

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Find market equilibrium….Find market equilibrium….

A t e q u i l i b r i u m 2142 QQ 123 Q S o e q u i l i b r i u m 4* Q T h u s e q u i l i b r i u m 624* P

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‘‘sketch’ the diagramsketch’ the diagram

Q Q* = 4

P

P*=6

0 Consumer Surplus

14 CS 14

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PS D

S

2

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Consumer surplus…Consumer surplus…

***

0QPdQQDCS

Q

i ) a r e a u n d e r e n t i r e d e m a n d c u r v e b e t w e e n Q = 0 a n d Q *

405616

01404144

14

142

22

4

02

4

0

QQ

dQQ

i i ) t o t a l r e v e n u e = a r e a u n d e r p r i c e l i n e a t P * = 6 , b e t w e e n Q = 0 a n d Q * = 4 i s P * Q * = 2 4

i i i ) S o C S = 4 0 – 2 4 = 1 6

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Producer Surplus…Producer Surplus…

*

0

** .Q

dQQSQPPS

i) area under Supply curve between Q = 0 and Q *

1688

0202

1424

2

1

22

1

2

22

4

0

2

4

0

QQ

dQQ

ii) total revenue = area under price line at P * = 6, between Q = 0 and Q * = 4 is P *Q * = 24

iii) So PS = 24 – 16 = 8

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Total SurplusTotal Surplus

• Total surplus = CS + PS = 16 + 8 = 24