chapter 6: integration

Chapter 6 Integration Chapter 6 Integration Nguyen Thi Minh Tam ntmtam vnuagmail com December 3, 2020 1 6 1 Indefinite integration 2 6 2 Definite integration 6 1 Indefinite integration Antiderivative A function F is called an anti derivative of f on an interval I if F ′(x) = f (x) ∀x ∈ I Example 1 F (x) = x5 is an anti derivative of f (x) = 5x4 Indefinite Integrals The indefinite integral of a function f (x), denoted ∫ f (x)dx, is defined by ∫ f (x)dx = F (x) + C, where F (x) is an anti derivat.

Chapter 6: Integration

Nguyen Thi Minh Tam

ntmtam.vnua@gmail.com

December 3, 2020

1 6.1 Indefinite integration

2 6.2 Definite integration

6.1 Indefinite integration

Antiderivative

A function F is called ananti-derivative of f on an interval I if

F0(x ) = f (x ) ∀x ∈ I

Example 1 F (x ) = x5 is an anti-derivative of f (x ) = 5x4

Indefinite Integrals

Theindefinite integralof a function f (x ), denoted

Z

f (x )dx , is defined by

Z

f (x )dx = F (x ) + C , where F (x ) is an anti-derivative of f (x )

Properties of Indefinite Integrals

1)

Z

kf (x )dx = k

Z

f (x )dx , k is a constant

2)

Z

[f (x ) + g (x )] dx =

Z

f (x )dx +

Z

g (x )dx

3)

Z

[f (x ) − g (x )] dx =

Z

f (x )dx −

Z

g (x )dx

4) If

Z

f (x )dx = F (x ) + C , then

Z

f (u)du = F (u) + C , where

u = u(x ) is a function of x

Basic Integration Formulas

1)

Z

kdx = kx + C

2)

Z

xndx = x

n+1

n + 1+ C (n 6= −1)

3)

Z

1

xdx = ln |x | + C

4)

Z

exdx = ex+ C

Z

emxdx = 1

me

mx+ C (m 6= 0)

Example 2 Find the indefinite integrals

a)

Z

(5x2− 3x + 2)dx

b)

Z 

7x3+ 4e−2x − 3

x2

 dx

Example 3 Find the indefinite integrals

a)

Z

(5x + 1)3dx

b)

Z

x (1 + x2)7dx

c)

Z 4x3

2 + x4dx

d)

Z

ex(1 + ex)3dx

Example 4.

a) Find the total cost if the marginal cost is

MC = 3e0.5Q and fixed costs are 10

b) The marginal revenue function of a monopolistic producer is

MR = 100 − 6Q Find the total revenue function and deduce the corresponding demand function

c) Find an expression for the savings function if the marginal propensity to save is given by

MPS = 0.4 −√0.1

Y and savings are zero when income is 100

6.2 Definite integration

Definite integrals

The definite integralof f from a to b is given by

Z b

a

f (x )dx = F (x )|ba = F (b) − F (a), where F (x ) is an anti-derivative of f (x )

If f is continuous on [a, b], f (x ) ≥ 0 for all x ∈ [a, b] then

Z b

a

f (x )dx is the area under the graph of f (x ) between x = a and x = b

6.2.1 Consumer’s surplus

Consumer surplusis the difference between the price that consumers are willing to pay and the actual price that they pay

Formula for computing the consumer’s surplus at Q = Q0

CS =

Z Q 0

0

f (Q)dQ − Q0P0, where P = f (Q) is the demand function

Example 5 Find the consumer’s surplus at Q = 8 for the demand function

P = 100 − Q

6.2.2 Producer’s surplus

Producer’s surplus is the difference between the market price and the price given by the supply curve

Formula for computing the producer’s surplus at Q = Q0

CS = Q0P0−

Z Q 0

0

g (Q)dQ, where P = g (Q) is the supply function

Example 6 Given the demand equation

P = 50 − 2QD and supply equation

P = 10 + 2QS calculate

a) the consumer’s surplus

b) the producer’s surplus

assuming pure competition

Note On the assumption of pure competition, the price is determined by the market Therefore, before we can calculate the producer’s surplus, we need to find the market equilibrium price and quantity

6.2.3 Investment flow

Net investment (I ) is defined to be the rate of change of capital stock (K ), so that

I = dK

dt , where

I (t): the flow of money, measured in dollars per year,

K (t): the amount of capital accumulated at time t, measured in dollars

Given the net investment function, the capital formation

during the time period from t = t1 to t = t2 is

Z t 2

t

I (t)dt

Example 7 The net investment function is given by

I (t) = 800t1/3 Calculate

a) the capital formation from the end of the first year to the end

of the eighth year,

b) the number of years required before the capital stock exceeds

$48 600

6.2.4 Discounting

If the discount rate is r , then the present value of a continuous revenue stream for n years at a constant rate of $S per year is computed by

P =

Z n

0

Se−rtdt

Example 8 Calculate the present value of a continuous revenue stream for 10 years at a constant rate of $5000 per year if the discount rate is 6%

Exercise 6.1, 6.1*3, 4, 6, 7 (page 496), 2, 7, 10, 12 (page 497)

Exercise 6.2, 6.2*4, 5, 7, 8, 9, 10 (page 509-510)