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Integration

Course Manual

Indefinite Integration 7.1-7.2

Definite Integration 7.3-7.4

Jacques (3rd Edition)

Indefinite Integration 6.1

Definite Integration 6.2 | |

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).

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

[pic]

Example

[pic]

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

How did we integrate f(x)?

Rule 1 of Integration:

[pic]

Examples

[pic]

check: if y = 1/3 x3 + c then dy/dx = x2

[pic]

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

Rule 2 of Integration:

[pic]

Examples

[pic]

[pic]

[pic]

Rule 3 of Integration:

[pic]

Example

[pic]

•Calculating Marginal Functions

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

Marginal Cost Function

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

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

[pic]

[pic]

[pic]

F = the constant of integration

If Q=0, then TC=F

F= Fixed Cost…..

Another Example

MC = f (Q) = Q + 5

If Total Cost = 20 when production is 0, find TC function?

[pic]

[pic]

[pic]

F = the constant of integration

If Q=0, then TC = F = Fixed Cost

So if TC = 20 then, [pic]

Another Example

Given Marginal Revenue, find the Total Revenue function

MR = f (Q) = 20 – 2Q

[pic]

[pic]

[pic]

c = the constant of integration

Example:

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

Solution:

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

2) Find TR and TC

[pic]

[pic]

[pic]

so TR = 22Q – Q2

MC = f (Q) = 2Q2 – 6Q + 6

[pic]

[pic]

[pic]

F = Fixed Cost = 0 (from question) so….[pic]

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

NOTE:

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

Definite Integration

The definite integral of f(x) between values a and b is:

[pic]

Example

[pic]

[pic]

The definite integral [pic] can be interpreted as the area bounded by the graph of f(x), the x-axis, and vertical lines x=a and x=b

The Consumer Surplus

Difference between value to consumers and to the market….

CS(Q) = oQ1ax - oQ1aP1

[pic]

Producer Surplus

Difference between market value and total cost to producers…

PS(Q) = oQ1aP1 - oQ1ay

[pic]

examples…..

Find a measure of consumer surplus

at Q = 5,

for the demand function p = 30 – 4Q

Solution

If Q = 5, then p = 30 – 4(5) = 10

[pic]

Entire area under demand curve between 0 and Q1 = 5:

[pic]

total revenue = area under price line (p1 = 10), between Q = 0 and Q1 = 5 is p1Q1

So CS = 100 – p1Q1 = 100 – (10*5) = 50

Example 2:

If p = 3 + Q2 is the supply curve, find a measure of producer surplus at Q = 4

Solution

If Q = 4, then p = 3 + 16 = 19

[pic]

Entire area under supply curve between Q = 0 and Q1 = 4…..

[pic]

total revenue = area under price line (p1 = 19), between Q = 0 and Q1 = 4 is p1Q1 = 76

So PS = p1Q1 – 331/3 =

76 – 331/3 = 422/3

Manual, Topic 7

Q3. A profit maximising firm has [pic] and [pic]. How much will it produce? What level of fixed costs would make the firm make zero profits?

Step 1: set MR=MC and find output that maximises profit, q*

[pic]

[pic]

Solve the quadratic for value of Q using formula [pic]:

a=1, b=-7, c=-8

[pic] so

[pic] (inadmissible) or [pic]

Thus 8 units produced by profit max firm

Step 2: integrate MR and MC to find TR & TC, and thus profits

[pic]

[pic]

In this case, the constant of integration [pic], since the firm makes no revenue when Q=0

[pic]

F, the constant of integration = Fixed Costs

[pic]

[pic]

Step 3: substitute in q* to TR and TC to get profit max values when producing q*

Substituting in [pic] for profit max. [pic]

Step 4: Set profit =0 (thus TR – TC = 0), & solve for F

Setting [pic], gives [pic]

Thus, value of F at (=0 is [pic]

Q4 (b): A firm which has no fixed costs has MC and MR given as follows:

MC=2Q2 – 6Q + 6;

MR = 22 – 2Q;

Find total profit for profit maximising firm when MR=MC?

Solution:

1) Find profit max output Q where MR = MC

22 – 2Q = 2Q2 – 6Q + 6

gives Q2 – 2Q – 8 = 0

Solve quadratic for Q, by using formula, or

(Q - 4)(Q + 8) = 0 so Q = +4 or Q =-2

so Q = +4 (since Q=-2 inadmissable)

2) Find TR and TC

[pic]

[pic]

[pic]

TR = c when Q=0; but TR = 0 when Q = 0; so therefore c = 0

so TR = 22Q – Q2

MC = f (Q) = 2Q2 – 6Q + 6

[pic]

[pic]

[pic]

F = Fixed Cost = 0 (from question) so….[pic]

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

so total profit when producing at MR=MC at q*=4 is

TR – TC = 72 - 182/3 = 53 1/3

Q5. The demand and supply functions for a good are given by the equations [pic] and [pic] respectively. Determine the equilibrium price and quantity and calculate the consumer and producer surplus at equilibrium.

At equilibrium

[pic]

[pic]

So equilibrium [pic]

Thus equilibrium [pic]

Consumer Surplus

Difference between value to consumers and to the market…. Area above price line and under Demand curve

[pic]

[pic]

[pic]

[pic]

[pic]

Producer Surplus

Difference between market value and total cost to producers… area below price line and above Supply curve

[pic]

[pic]

[pic]

[pic]

[pic]

Total Surplus = CS + PS = 16 + 8 = 24

-----------------------

Producer Surplus

Supply Curve:

P = g(Q)

P

y

Consumer Surplus

x

Q

Q1

a

0

P1

P

f(x)

Demand Curve:

P = f(Q)

[pic]

[pic]

Q1

0

P1

a

[pic]

[pic]

b

Q

a

x

30

P

Demand Curve:

P = f(Q) = 30 – 4Q

P1=10

Consumer Surplus

Q1 = 5

7.5

0

Q

Supply Curve:

P = g(Q) = 3 + Q2

P

P1 = 19

Producer Surplus

3

Q1 = 4

0

Q

S

CS[pic]

P

14

P*=6

2

PS

D

Q* = 4

7

0[pic]

Q

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