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)
Example
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:
Examples
check: if y = 1/3 x3 + c then dy/dx = x2
check: if y = x + c then dy/dx = 1
Rule 2 of Integration:
Examples
Rule 3 of Integration:
Example
•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
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?
F = the constant of integration
If Q=0, then TC = F = Fixed Cost
So if TC = 20 then,
Another Example
Given Marginal Revenue, find the Total Revenue function
MR = f (Q) = 20 – 2Q
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
so TR = 22Q – Q2
MC = f (Q) = 2Q2 – 6Q + 6
F = Fixed Cost = 0 (from question) so….
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:
Example
The definite integral 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
Producer Surplus
Difference between market value and total cost to producers…
PS(Q) = oQ1aP1 - oQ1ay
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
Entire area under demand curve between 0 and Q1 = 5:
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
Entire area under supply curve between Q = 0 and Q1 = 4…..
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 and . 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*
Solve the quadratic for value of Q using formula :
a=1, b=-7, c=-8
so
(inadmissible) or
Thus 8 units produced by profit max firm
Step 2: integrate MR and MC to find TR & TC, and thus profits
In this case, the constant of integration , since the firm makes no revenue when Q=0
F, the constant of integration = Fixed Costs
Step 3: substitute in q* to TR and TC to get profit max values when producing q*
Substituting in for profit max.
Step 4: Set profit =0 (thus TR – TC = 0), & solve for F
Setting , gives
Thus, value of F at =0 is
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
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
F = Fixed Cost = 0 (from question) so….
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 and respectively. Determine the equilibrium price and quantity and calculate the consumer and producer surplus at equilibrium.
At equilibrium
So equilibrium
Thus equilibrium
Consumer Surplus
Difference between value to consumers and to the market…. Area above price line and under Demand curve
Producer Surplus
Difference between market value and total cost to producers… area below price line and above Supply curve
Total Surplus = CS + PS = 16 + 8 = 24