Introduction to Mathematical Economics Lecture 11

  Introduction to Mathematical Economics Lecture 11

  Ekki Syamsulhakim MApplEc

  

Undergraduate Program Faculty of Economics & Business Universitas Padjadjaran Today: Integral Calculus

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
• – functions and see how this can be used in Economics.

  

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
• derivative of a function you will end up with the same function that you started with (be careful with constants however)

  Integration as the reverse of

differentiation

If you have:

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

  have y = x

• – So the reverse of differentiation must have a
• – constant, example: if dy/dx = 2x, then

  2

  y = x +C

  

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

  f ( x ) dx 

  

The indefinite integral

  Inverse operation of differentiation 

• antidifferentiation. Antiderivative  indefinite integral.
• � ( � ) ��=� ( � ) �

  ∫

Geometric Interpretation

  Derivative: slope of a curve

• Integral: the area beneath a curve
• between two points  the definite integral.

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
• shapes, by the way?

  

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 x and x ); How do you calculate it?

  1 y y=a x x x ₁

  1

  

Finding the area under a curve

• x ) or = a. Dx
• The area is = a.(x

  x x ₁ y x y=a

  Dx a

  

Finding the area under a curve

One could, for instance, divide the length

• of x x and add up the area of the two

  1 boxes created, and still get the same value of the area y y=a a x x x ₁ Dx Dx

  

Finding the area under a curve

The area now is = a. Dx +a. Dx

• 1

  2 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

  y y=a a x x x ₁ Dx Dx

  

Finding the area under a curve

With smaller boxes, the “ Dx”s would be

• smaller (approaching zero), but the number of boxes increases, as one will have many boxes at the same time

  y y=a a x x x ₁ Dx

  Dx

  

Finding the area under a curve

Using mathematical notation SIGMA ( S),

• which means “add up”, we can write the

  area as

  4 a x a x a x a x a x

           i

  1

  2

  3

  4  i

  1  y y=a a x x x ₁ Dx

  Dx

  

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?

  y x a b

  

Finding the area under a curve

• You can use similar method, by dividing the area into boxes, as follows:

  y a x

  4

  4

  3

  3

  2

  

2

  1

  1

  4

  1 x y x y x y x y x y i i i

            _

  y a b x  ^ _ ^{ b x} _{a x} ^{x y}

  

Finding the area under a curve

• But then, you notice that the calculation is NOT accurate (see the small “triangles”)

  Dx”s

  

Finding the area under a curve

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

  y a b x  ^ _ ^{ b x} _{a x} ^{x y}

  y a b x  ^ _ ^{ b x} _{a x} ^{x y}

  

Finding the area under a curve

• More boxes you get, smaller “triangles” you have, so the approximation is better than before!

  

Definite Integration and Applications

of Integrals in Economics

We can use the S symbol together with the limit

• notation to describe the situation as x gets smaller and smaller,
• x b

   x b b

   i

  4 

lim y x y dx y dx

   i i

   x

      x a

   x a a

   i

  1 

  

Definite Integration and Applications

of Integrals in Economics

Notice the transformation of the mathematical

• symbol, the combination of limit operator together

  with the S symbol to the elongated s symbol

  x b  x b b

   i

  4 

lim y x y dx y dx

   i i

   x

      x a

   x a a

   i

  1 

  

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 gets smaller the area is more precise.

  

Rules of integration

• There are at least 24 rules of integration

NOTE:

• – (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
• – later

  

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

• – integrate the integrand

  ∫ �

  

Rules of integration

• Rule 1: Power rule

  � ��= �

  �+1 �+1 +� �≠− 1

  

Rules of integration

Rule 1’:

• Generalised “power rule”/ substitution rule

  n

  1  f x ( ) n

    f x ( ) f x dx '( )

  C n

  1      

   n

  1 

  

Rules of integration

• Hence, Rule 1’
• C

     

   ( 1 )] ' [ )] ( [

  )] ( [

  1 n x f x f dx x f n n Examples 3 dx 3 x c  

  

  1

  2 x dx x c

    

  2

  4 (2 x 2)

  3  (2 x 2) (2) dx C   

  

  

4

  4

  ( 2 x 2 )

  3 

  ( 2 x 2 )   + C

  

  8

  

Rules of integration

Rule 2 : Integral of

• a sum:

  Rule 3 : Integral of

• a constant multiple:

Examples

  3

                 

               

  3 x dx dx xdx x x c xdx xdx x x c x x dx xdx x dx x x c

  2

  2

  3

  2 (3 2 )

  3

  2

  2

  2

  2

  3

  3

  1

  3

  2

  3

  3 (3 2 )

  2

  2

  2

  2

  3

  

Rules of integration

Rule 4 : Exponential

• rule:

  Rule 4’: Generalised

• Exponential rule:

  

Rules of integration

Rule 4’: Generalised Exponential rule:

• f ( x )

  e f ( x ) e dx C

     f ' ( x )

Examples

  4

  4

  2

  5

  2

  5

  2

  5

  1

  2

  2 x x x e dx e c e e dx c e c

    

       

   

  

Rules of integration

Rule 5 : “power of -1” rule:

• Rule 5’: Generalised
• “power of -1” rule :

  

Examples

  1

  1 ln

  1

  2 2 2.ln dx x C x

x dx dx x c

x

  

      

    

  

Rules for logarithm function

ln x dx x (ln x 1 ) c   

   ( mx b )[ln( mx b ) 1 ]    ln( mx b ) dx c

    

   m

  

Example

  Integration by parts

• Consider two continuous functions u=f(x) and v=g(x), then,

  [ ( ). ( )] ( ). '( ) '( ). ( ) ( ) ( ) ( ) . .

  . .

  . . Let us assume that and Then, Let us integrate both sides Rearranging: d f x g x f x g x f x g x dx u f x v g x d uv u dv du v uv u dv v du u dv uv v du

           

   

  

Examples

Using integration by parts, find 3 dx

   u 3, dv dx

    du 0, v x

    u dv uv . v du .

     

  3. dx 3 x x .0 3 x c      

  Examples

  Using integration by parts, find (2 x 3)(2 ) x 

   u (2 x 3), dv

  2 x   

  2 du 2, v

  2 x x   

   u dv uv . v du .

   

   

  2

  2

  (2 x 3).(2 ) (2 x x 3)( ) x x .2    

   

  3

  2

  2

  (2 x 3 ) 2 x x   

  

  2

  4

  3

  2

  3

  3

  2

  2 x 3 x x x 3 x c      

  3

  3

Examples

  4

                

  3 Verification x x x x dx x dx x dx x x c x x c

  2

  3

  3

  6

  2

  2

  6

  4

  2 (2 3)(2 ) (4 6 )

  3

  2

  3

  4

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 points (example: x=a and x=b)

  dx ) x ( dF ) x ( f

  

How to calculate definite

integrals

• The fundamental theorem of calculus says

  ) a ( F ) b ( F ) x ( F dx ) x ( f a b b a

       Where 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

  8 dx x  

  2

  

Properties of definite integrals

  1. Reversing the order of the limits of integration changes the sign of the definite integral.

   

 

a b b a

  (x) dx f dx f (x)

 

• – Example:

   

  1

  2

  2

  2

  1

  2

  3

  3 dx x dx x

  

Properties of definite integrals

  2. If the upper limit of the integration equals

the lower limit, the value of the definite

integral is zero.

     

  ) a ( F ) a ( F dx ) x ( f a a

• – Example:

  4x)

  3 ⁷ ₇ ²   

  ( dx x

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

• – as long as a

  bc

       c b b a c a dx f dx f dx f

  (x) (x) (x)

  

• – Example:

   

  5

  2

  2

  5 8 x 8 x

  8 dx x dx dx

  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 b b b f (x) dx g (x) dx f (x) g (x) dx

      

     a a a

  Example:

  2

  2

  2 4 x dx 2 xdx 6 x dx  

    

  1

  1

  1

  

5. The definite integral of a constant times a

function is equal to the constant times the definite integral.

  (x) (x)  

  



b a b a dx f k dx kf dx dx b a x 6 x

• – Example:

  6

  2

  1  

  

  

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

  dTC MC TC MC dQ

  

   dQ

• – 2

  Example: MC=Q +2Q+4 Find the total cost if FC=100

  2Q 4) dQ

    From Marginal Revenue to Total

• Revenue to the Demand Curve:

  dTR MR TR MR dQ   

   dQ

  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.
• Consumers’ surplus is the (sum of) the utility that

Consumers’ surplus:

• – consumers received but is not paid for Measured by the area under Demand Function – p

  P q q

• Producers’ surplus is the sum over all units

Producers’ surplus:

• – produced by a firm of differences between market

  price of a good and marginal costs of production

  Measured by the area above Supply Function – p

  P q q