Introduction to Mathematical Economics Lecture 11

Introduction to Mathematical Economics

Lecture 11

Ekki Syamsulhakim MApplEc https://sites.google.com/a/fe.unpad.ac.id/ekki/ ekki.syamsulhakim@fe.unpad.ac.id

2

3

Learning outcomes

– Understand what is meant by “integration”

and carry out the procedure confidently.

– _{Understand the difference between definite}

and indefinite integrals.

– _{Use the method of integration to integrate}

4

Integration as the reverse of

differentiation

• _{There are many mathematical operations}

which are the reverse of each other.

• _{Multiplication and division} • _{Logs and antilogs}

• _{Integration and differentiation}

• _{If we integrate a derivative or take the}

5

Integration as the reverse of

differentiation

• _{If you have:}

– dy/dx = 2x, then by trial and error, you will

have y = x2

– _{But if you have y = x}2+n, dy/dx is also = 2x

– So the reverse of differentiation must have a

6

Symbolization

• The integration symbol is ; (the

elongated s), refers to a summation

• If we want to integrate the function y=f(x)

w.r.t. x we write



7

The indefinite integral

• Inverse operation of differentiation _

antidifferentiation.

• _{Antiderivative  indefinite integral.}

8

Geometric Interpretation

• _{Derivative: slope of a curve}

• Integral: the area beneath a curve

9

Finding the area under a curve

• _{Last time we saw that integration is the}

reverse of differentiation

• _{Now we will see how integration can give}

us the area under a curve

• _{How would you calculate the area of these}

10

Finding the area under a curve

• _{Now, suppose that you have a constant}

function y=a, and you intend to calculate the area below the curve of y=a (between x₀ and x₁); How do you calculate it?

x₀ x₁

y

x y=a

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Finding the area under a curve

• The area is = a.(x₁-x₀) or = a.Dx

x₀ x₁

y

x y=a

0

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Finding the area under a curve

• _{One could, for instance, divide the length}

of x₀x₁ and add up the area of the two

boxes created, and still get the same value of the area

x₀ x₁

y

x y=a

0

Dx₁ a

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Finding the area under a curve

• The area now is = a.Dx₁+a.Dx₂

• One could now divide the two boxes into

another two, and add up the area of the four boxes and still get the same value

x₀ x₁

y

x y=a

0

Dx₁ a

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Finding the area under a curve

• _{With smaller boxes, the “}D_{x”s would be}

smaller (approaching zero), but the

number of boxes increases, as one will have many boxes at the same time

x₀ x₁

y

x y=a

0

Dx₁ a

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Finding the area under a curve

• _{Using mathematical notation SIGMA (}S_),

which means “add up”, we can write the area as

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Finding the area under a curve

• _{What if we don’t have a constant function,}

instead we have y=f(x), say a cubic

function as shown below; How would you calculate the area?

a _b

y

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Finding the area under a curve

• You can use similar method, by dividing the area

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Finding the area under a curve

• But then, you notice that the calculation is NOT

accurate (see the small “triangles”) y

a _b x

  



b x

a x

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Finding the area under a curve

• Try to divide the area in more boxes, or get

smaller “Dx”s y

a _b x

  



b x

a x

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Finding the area under a curve

• More boxes you get, smaller “triangles” you

have, so the approximation is better than before! y

a _b x

  



b x

a x

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Definite Integration and Applications

of Integrals in Economics

• _{We can use the} S _{symbol together with the limit}

notation to describe the situation

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Definite Integration and Applications

of Integrals in Economics

• _{Notice the transformation of the mathematical}

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Definite Integration and Applications

of Integrals in Economics

• The most important (economic) applications of

integration are to find the area under a curve between two points x=a and x=b.

– _{As such there are is no geometric formula to find}

the area under an irregular curve y=f(x) but,

– _{we can approximate this area by subdividing the}

interval [a,b] into n subintervals and creating

rectangles such that the height of each rectangle is equal to the smallest value of the function in the subinterval. This is not very precise but as x

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Rules of integration

• NOTE:

– There are at least 24 rules of integration

(summarized in “integrals table” or “table of integrals)

• Most of them is found by relating the rule to the

associated derivation rule, or by expanding the “basic rules of integrals”

– We will discuss some of the important rules – You might have to study the integrals table

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Rules of integration

• _{The integral of zero is a constant}

• The integral of a constant function, f(x) =

k, is k times the variable

– _{Don’t forget to add a constant after you}

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Rules of integration

• Rule 1: Power rule

∫

�

�

��

=

�

�+1

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Rules of integration

• Rule 1’:

Generalised “power rule”/ substitution rule









1

( )

( )

'( )

1

1

n

n

f x

f x

f x dx

C n

n







 



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Rules of integration

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Rules of integration

• _{Rule 2 : Integral of}

a sum:

• _{Rule 3 : Integral of}

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Rules of integration

• Rule 4 : Exponential

rule:

• _{Rule 4’: Generalised}

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Rules of integration

• Rule 4’: Generalised Exponential rule:

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Examples

4 4

2 5

2 5

1

2 5

2

2

x

x x

e dx e

c

e

e

dx

c

e

c



 







 





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Rules of integration

• Rule 5 : “power of -1” rule:

• _{Rule 5’: Generalised}

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Examples

1

1

ln

1

2

2

2.ln

dx

x C

x

x dx

dx

x c

x















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Rules for logarithm function



ln

x

dx



x

(ln

x



1

)



c















c

m

b

mx

b

mx

dx

b

mx

)

(

)[ln(

)

1

]

Example

•

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Integration by parts

• _{Consider two continuous functions u=f(x) and v=g(x),}

then,

Let us integrate both sides

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Using integration by parts, find

dx

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Using integration by parts, find x x

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Examples

2

2 3 2 3 2

(2 3)(2 ) (4 6 ) 4 6 4

4 6 3

3 2 3

Verification x x x x dx

x dx x dx x x c x x c

  

       





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Definit Integrals Once Again

• _{We have discussed that the most important}

(economic) applications of integration are to find the area under a curve between two

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How to calculate definite

integrals

• _{The fundamental theorem of calculus says}

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How to calculate definite

integrals

• _{The integral between limits is known as}

the definite integral of f(x) from a to b, where a is the lower limit and b is the upper limit

• _Example:





6

2

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Properties of definite integrals

1. Reversing the order of the limits of

integration changes the sign of the definite integral.

– _Example:

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Properties of definite integrals

2. If the upper limit of the integration equals the lower limit, the value of the definite integral is zero.

– _Example:

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3.The definite integral can be expressed as the sum of component sub-integrals

– _{as long as a}_b_c – Example:

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4.The sum or difference of two definite integrals with identical limits of

integration is the integral of the sum or difference of the two functions

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5. The definite integral of a constant times a function is equal to the constant times the definite integral.

– _Example:

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Applications of Integration

• From marginal cost to total cost

– _{Suppose we are faced with a MC function.}

We can use (indefinite) integration to find the TC function

– _{Example: MC=Q}2+2Q+4 Find the total cost if FC=100



 

 TC MC dQ

dQ dTC MC









dQ

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• _{From Marginal Revenue to Total}

Revenue to the Demand Curve:

– _{To find an expression for total revenue}

from any given demand equation, we normally multiply by Q as TR=PQ so P=TR/Q (the inverse demand curve)

– _{Example: MR=10-4Q find TR and the}

demand curve.









TR

MR

dQ

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• Consumers’ surplus:

– Consumers’ surplus is the (sum of) the utility that

consumers received but is not paid for

– Measured by the area under Demand Function

P₀

q₀

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• Producers’ surplus:

– Producers’ surplus is the sum over all units

produced by a firm of differences between market price of a good and marginal costs of production

– Measured by the area above Supply Function

P₀

q p