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Integral Calculus - Lecture Notes | MATH 205, Study notes of Calculus

Wellesley CollegeCalculus

Material Type: Notes; Class: Multivariable Calculus; Subject: Mathematics; University: Wellesley College; Term: Unknown 1989;

Typology: Study notes

Pre 2010

Uploaded on 08/18/2009

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Fall Semester ’07-’08
Akila Weerapana
LECTURE 5: INTEGRAL CALCULUS
I. INTRODUCTION
•In the last lecture, we talked about differentials and used differentials to talk about some
important economic concepts like, elasticities, economic growth and simple comparative static
analysis. Since Econ 205 is a pre-requisite, that will be all we will cover on differential calculus
theory. You should be very comfortable with any other topic related to differential calculus
at the level of Math 205.
•Today’s lecture does a similar review of concepts regarding integral calculus from Math 205.
Once again, we will do a quick review and spend most of the time discussing economic
applications. The most important applications will be calculating consumer and producer
surplus, bond pricing and some econometric applications with density functions.
•If you need to brush up on your integrals, once again I urge you to do so immediately. Chapter
12 of Klein will serve both as a review of what you studied in your calculus classes and as
supplementary reading for this lecture.
II. THEORY
•The indefinite integral F(x) = Rf(x)dx is the (family of) anti-derivative(s) of a function.
In other words, dF (x)
dx =f(x).
•Since constant terms have a derivative of zero, we can only define F(x) up to an arbitrary
constant absent any further information, hence the family of anti-derivatives.
•In essence, you can think of integration as the reverse of differentiation. Suppose that we
have a function f(x), that is the derivative of some function F(x). Integration can be used
(with a little bit of additional information to uncover the constant terms) to recover the
original function F(x) using the information contained in f(x).
•The definite integral Rb
af(x)dx is the area under the curve f(x) over the range x=ato
x=b.
•The value of Rb
af(x)dx =F(b)−F(a) where F(x) = Rf(x)dx
•In Economics, this means that we can derive a total cost function using information about
marginal costs, derive the underlying utility function given information about marginal util-
ity and derive a cumulative density function given information about a probability density
function, etc.
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Partial preview of the text

Download Integral Calculus - Lecture Notes | MATH 205 and more Study notes Calculus in PDF only on Docsity!

Fall Semester ’07-’ Akila Weerapana

LECTURE 5: INTEGRAL CALCULUS

I. INTRODUCTION

  • In the last lecture, we talked about differentials and used differentials to talk about some important economic concepts like, elasticities, economic growth and simple comparative static analysis. Since Econ 205 is a pre-requisite, that will be all we will cover on differential calculus theory. You should be very comfortable with any other topic related to differential calculus at the level of Math 205.
  • Today’s lecture does a similar review of concepts regarding integral calculus from Math 205. Once again, we will do a quick review and spend most of the time discussing economic applications. The most important applications will be calculating consumer and producer surplus, bond pricing and some econometric applications with density functions.
  • If you need to brush up on your integrals, once again I urge you to do so immediately. Chapter 12 of Klein will serve both as a review of what you studied in your calculus classes and as supplementary reading for this lecture.

II. THEORY

  • The indefinite integral F (x) =

f (x)dx is the (family of) anti-derivative(s) of a function. In other words, dF dx^ (x )= f (x).

  • Since constant terms have a derivative of zero, we can only define F (x) up to an arbitrary constant absent any further information, hence the family of anti-derivatives.
  • In essence, you can think of integration as the reverse of differentiation. Suppose that we have a function f (x), that is the derivative of some function F (x). Integration can be used (with a little bit of additional information to uncover the constant terms) to recover the original function F (x) using the information contained in f (x).
  • The definite integral

∫ (^) b a f^ (x)dx^ is the area under the curve f(x) over the range^ x^ =^ a^ to x = b.

  • The value of

∫ (^) b a f^ (x)dx^ =^ F^ (b)^ −^ F^ (a) where^ F^ (x) =^

f (x)dx

  • In Economics, this means that we can derive a total cost function using information about marginal costs, derive the underlying utility function given information about marginal util- ity and derive a cumulative density function given information about a probability density function, etc.

Basic Rules of Indefinite Integrals

  • Since almost everyone uses integration less than differentiation in academic settings, it may be wise to do a quick survey of some basic rules of integration. Some useful rules to remember (where a is an arbitrary constant term) are ∫ xndx =

xn+ n + 1

  • c

adx = ax + c ∫ exdx = ex^ + c

x dx^ = ln(|x|) +^ c ∫ [f (x) + g(x)]dx =

f (x)dx +

g(x)dx ∫ af (x)dx = a

f (x)dx

  • Note that c is an arbitrary constant. Since constants disappear during differentiation without additional information we can not identify the constant terms in the original function by integration, i.e. given f ′(x) = 2x we can not state whether f (x) = x^2 + 3 or f (x) = x^2 + 5
  • There are also two important rules of substitution that you should be familiar with.
    1. If u=g(x) then

f (u)g′(x)dx =

f (u)du

udv = uv − v

du

Examples:

  1. Find the value of the following integral

∫ (^2) x x^2 +3 dx.^ Let^ u^ =^ g(x) =^ x

(^2) + 3. Then du = g′(x)dx = 2xdx. We can then rewrite the integral as

u du^ the solution to which is ln(|u|) + c. So (^) ∫ 2 x x^2 + 3

dx = ln(x^2 + 3) + c

  1. Find the value of the following integral

ln(x)dx. Let u = ln(x) and v = x. Then du = (1/x)dx and dv = dx. Since we have an integral of the form

udv we have a solution of the form uv −

vdu = x ln(x) −

x

x dx

= xln(x) −

dx so ∫ ln(x)dx = x ln(x) − x + c

Basic Rules of Definite Integrals

  • Other important rules to keep in mind are ∫ (^) a

a

f (x)dx = 0 ∫ (^) b

a

f (x)dx = −

∫ (^) a

b

f (x)dx ∫ (^) c

a

f (x)dx =

∫ (^) b

a

f (x)dx +

∫ (^) c

b

f (x)dx where a < b < c

CS =

∫ Q∗

0

[

D−^1 (Q) − P ∗

]

dQ

or equivalently CS =

∫ P¯

P ∗

[D(P )] dP

  • Producer surplus is the area between P ∗^ and the supply curve between Q = 0 and Q = Q∗.

P S =

∫ Q∗

0

[

P ∗^ − S−^1 (Q)

]

dQ

or equivalently P S =

∫ P ∗

P

[S(P )] dP

Example:

  • Suppose the demand curve for a product is given by D(P ) = 10 − 2 P and the supply curve is given by S(P ) = 4 + P. Equilibrium price and quantity can be calculated as P ∗^ = 2 and Q∗^ = 6. The inverse demand and supply functions can be calculated as D−^1 (Q) = 5 − Q 2 and S−^1 (Q) = Q − 4
  • The y-intercepts are at P¯ = 5 for demand and P = −4 for supply.
  • Using integrals, we can calculate consumer surplus to be

CS =

0

[

D−^1 (Q) − 2

]

dQ

0

[(

Q

]

dQ =

0

[

Q

]

dQ

3 Q −

Q^2

6

0

  • Alternatively, we could have found consumer surplus as

CS =

2

[D(P )] dP

2

[10 − 2 P ] dP

=

10 P − P 2

2 = (50^ −^ 25)^ −^ (20^ −^ 4) = 9

  • Using integrals, we can calculate producer surplus to be

P S =

0

[

2 − S−^1 (Q)

]

dQ

0

[2 − (Q − 4)] dQ =

0

[6 − Q] dQ

6 Q −

Q^2

6

0

  • Alternatively, we could have found producer surplus as

P S =

− 4

[S(P )] dP

− 4

[4 + P ] dP

4 P +

P 2

2

− 4

  • Of course, since the curves are linear, we can avoid all of this and calculate the magnitudes of consumer and producer surplus geometrically. Consumer surplus is the area covered by a right-angled triangle with base 6 and height 3 so CS = 1/ 2 ∗ 6 ∗ 3 = 9. Producer surplus is the area under a right-angled triangle with base 6 and height 6 so P S = 1/ 2 ∗ 6 ∗ 6 = 18.

Welfare Effects of Price Changes

  • We can also use integrals to think about the welfare effects of changes in equilibrium price and quantity. For example: what happens to consumer and producer surplus when price goes up? Graphically, let’s think about what happens when there is an increase in demand that raises equilibrium price to P ∗∗^ and equilibrium quantity to Q∗∗. -

6

P

Q

CS

PS

Q∗∗

P ∗∗

D′−^1 (Q)

S−^1 (Q)

D−^1 (Q)

H HH HH HH HH HH HH

HH HH HH HH HH HH HH

  • New consumer surplus is the area under the new demand curve above P ∗∗^ and the new producer surplus is the area under the supply curve below P ∗∗.

Example:

6

Proportion of population

Cumulative share of income

A

B

100% 45-degree line

L(p) [Lorenz Curve]

Random Variables

  • We can also use integrals to think about basic concepts related to econometrics and statistics. In statistics, we often use random variables, variables whose value reflect the outcome of some probabilistic event.
  • Random variables have a probability distribution, which describes the values that the random variable can take on, and the probability of achieving each of those outcomes. For example, suppose X is a random variable whose value, x, is the outcome from a single roll of a 6 sided fair die. The probability distribution of X, f(x) is f(1)=1/6, f(2)=1/6, f(3)=1/6, f(4)=1/6, f(5)=1/6, f(6)=1/6 and f(i)=0 for any other value i.
  • For any discrete random variable X with the probability distribution function f(x), the fol- lowing rules must hold 0 ≤ f (x) ≤ 1 and

∀xi f^ (xi) = 1 i.e. the probability of observing any outcome has to be non-negative and the sum of the probability of observing the independent outcomes can’t exceed 1.

  • Random numbers can be continuous as well as discrete. Suppose that Z is a continuous random variable that can take on a continuum of values z. Since z can take on an infinite number of values, we can’t describe the probability of taking any given value: we can only talk about the probability that z taking on a range of values as P r(a ≤ z ≤ b) =

∫ (^) b a f^ (z)dz^ ≥^0

  • As in the discrete case, the second condition states that all the probabilities have to add up to 1: so

−∞ f^ (z)dz^ = 1. Note that^ f^ (i) = 0 if Z does not take on the value^ i.

  • Possible distribution functions for a discrete random variable X and a continuous random variable Z, are given below.

6

f (x)

x

1 6

1 2 3 4 5 6

6

f (z)

z

  • Other key concepts associated with random variables include the cumulative distribution function F (x) defined as

F (x) = P rob[X ≤ x] =

∫ (^) x

−∞

f (x)dx

  • The expected value of a random variables defined as

μ = E(X) =

−∞

xf (x)dx

  • The variance of a random variable defined as

E(x − μ)^2 = E(x^2 − 2 μx + μ^2 ) = E(x^2 ) − 2 μ(E(x)) + μ^2 = E(x^2 ) − μ^2

  • This can be calculated as

V ar(x) =

−∞

x^2 f (x)dx − μ^2

Examples

  • Let’s do some examples with a couple of commonly used probability distributions highlighted by Klein: the uniform distribution and the exponential distribution.
  • The uniform distribution has an upper limit (b) and a lower limit (a) and has a distribution function

f (x) =

b − a for x in [a, b] f (x) = 0 otherwise

  • The expected value of a random variable that is exponentially distributed is

E(x) =

−∞

xf (x)dx =

0

[

xλe−λx

]

dx

Define u = x, dv = λe−λxdx then du = dx, v = −e−λx E(x) = uv −

vdu = −xe−λx

0

0

[

−e−λx

]

dx

= −xe−λx

0

e−λx λ

∞

0 = −

x +

λ

e−λx

∞ 0

= −

x +

λ

e−λx

0

λ

  • The variance of a random variable that is exponentially distributed is

V ar(x) = E(x − μ)^2 = E(x^2 ) − μ^2

=

−∞

x^2 f (x)dx − μ^2

We can calculate the value of the first term in the above expression as

Define u = x^2 , dv = λe−λxdx then

∫ du^ = 2xdx, v^ =^ −e−λx ∞ 0

[

x^2 λe−λx

]

dx = uv −

vdu = −x^2 e−λx

∞ 0

0

[

−e−λx

]

(2x)dx

= −x^2 e−λx

∞ 0

λ

0

[

λxe−λx

]

dx

= −x^2 e−λx

∞ 0

λ

λ

0

[

x^2 λe−λx

]

dx =

λ^2

Therefore

V ar(x) =

λ^2

λ^2

λ^2