Lecture six

1
Lecture six

• Integration

2
The definite integral definition

• The area A under a function f(x) is given by
• The expression is
  called the definite integral of
• function yf(x)

3
The definite integral definition cntnd.

• Diagramatically

error
y
y
y
error
yf(x)
yf(x)
f(x2)
f(x1)
A3
f(x1)
A
A2
x
x
x
x1
x2
x1
x2
x3
Area A1f(x1)(x2-x1)
A2A3f(x1)(x2-x1)f(x2)(x3-x2)
4
The indefinite integral

• The definite integral is a
  definite number
• (an area under a curve), which depends on the
  values a and b.
• If the upper limit b is not specified, but is
  variable, then we have an indefinite integral
  which is written as

y
x
x variable
a
5
Integration is the opposite of differentiation

y
yf(x)
x
a
x
x?x
6
Rules of integration

• The power rule

7
Rules of integration cntnd.

• Logarithmic rule
• Exponential rule

8
Rules of integration cntnd.

• Scalar multiplication rule
• The integral of a sum and difference

9
Rules of integration cntnd.

• Example 1
• Example 2

10
Rules of integration cntnd.

• Substitution rule
• This is the integral counterpart of the chain
  rule, namely

11
Rules of integration cntnd.

• Example

12
Rules of integration cntnd.

• Examples

13
Additional rules for definite integrals

• As we saw definite integrals are evaluated
  between two points in the domain of the function
  f(x) upper limit (b) and lower limit (a).

14
Additional rules for definite integrals cntnd.

• The calculation of definite integrals proceeds
  in two steps
• Step 1 Find the primitive function F(x). Note
  that the arbitrary constant of integration may be
  skipped since it will drop out in the next step.
• Step 2 Substitute xa to find F(a) and xb to
  find F(b) and calculate their difference
  F(b)-F(a)

15
Additional rules for definite integrals cntnd.

• Example
• Step 1
• Step 2

16
Properties of definite integrals
17
Economic applications

• Example 1 Deriving a total cost function from a
  marginal cost function (a) Given MC3q5, find
  the Total Cost function, (b) For the same
  marginal cost function, show that the variable
  cost of producing 10 units of output can be
  measured by the appropriate area under the MC
  curve.

18
Economic applications cntnd.

• Solution

TC
TVC
10
q
MC
MC3q5
q
0
10
19
Economic applications cntnd.

• Example 2 Deriving total revenue from the
  marginal revenue function Given MR100-2q, find
  the total revenue function and the demand
  function. Show the graphs of p, MR, TR for q40.

20
Economic applications cntnd.

• Solution

P, MR
q
MR
p
TR
2400
q
40
21
Economic applications cntnd.

• Example 3 Consumers surplus Find the
  consumers surplus for inverse demand function
  p100-0.5q and price p020.

22
Economic applications cntnd.

• Solution

p
Consumer surplus
A
p0
B
q
q0
23
Economic applications cntnd.

• Example 4 Producers surplus Find the
  producers surplus for inverse supply function
  p3q5.

24
Economic applications cntnd.

• Solution

Producer surplus
p
p3q5
PSp0q0-
p0
q0
q