1
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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