Chapter 5Integration

5.1 Anti differentiation: The Indefinite Integral

Anti differentiation

A function F(x) is said to be an antiderivative of f (x) if F'(x) = f (x) for every x in the domain of f(x).

The process of finding antiderivatives is called anti differentiation or indefinite integration.

Example.

Integral

General Antiderivative

Antiderivative Rules

• The constant rule

• The power rule

• The logarithmic rule

• The exponential rule 𝑒𝑘𝑥𝑑𝑥 =1

𝑘𝑒𝑘𝑥 + 𝐶, 𝑘 ≠ 0

More Antiderivative Rules

Notice that the logarithm rule “fills the gap” in the power rule; namely, the case where n=-1. You may wish to blend the two rules into this combined form:

න𝑥𝑛𝑑𝑥 = ൞𝑥𝑛+1

𝑛 + 1+ 𝐶, 𝑛 ≠ −1

ln 𝑥 + 𝐶, 𝑛 = −1

Examples

Find these integrals:

1. 3𝑑𝑥

2. 𝑥17𝑑𝑥

3. 1

𝑥𝑑𝑥

4. e−3 x dx

5. 𝑥3+2𝑥−7

𝑥𝑑𝑥

5.2 Integration bySubstitution

Integration by Substitution

How to do the following integral?

Think of u=u(x) as a change of variable whose differential is

Substitution Steps

Examples

1. Find

2. .

3. .

4. .

5. .

More Examples

The price p (dollars) of each unit of a particular commodity is estimated to be changing at the rate

𝑑𝑝

𝑑𝑥=

−135𝑥

9+𝑥2,

where x (hundred) units is the consumer demand (the number of units purchased at that price). Suppose 400units are demanded when the price is $30 per unit.

a. Find the demand function p(x)

b. At what price will 300 units be demanded? At what price will no units be demanded?

c. How many units are demanded at a price of $20 perunit?

5.3 The Definite Integral

Definite Integral

Our goal in this section is to show how area under a curve can be expressed as a limit of a sum of terms called a definite integral.

Area Under a Curve

Let f(x) be continuous and satisfy f(x)≥0 on the interval a≤x≤b. The region under the curve y=f(x) over the interval a≤x≤b has area𝐴 = lim

𝑛→∞𝑆𝑛 = lim

𝑛→∞𝑓(𝑥1) + 𝑓(𝑥2) + ⋯+ 𝑓(𝑥𝑛) ∆𝑥

where 𝑥𝑖 is the point chosen from the i-th subinterval if theinterval a≤x≤b is divided into n equal parts, each of length ∆𝑥 =

𝑏−𝑎

𝑛

The sum 𝑓(𝑥1) + 𝑓(𝑥2) + ⋯+ 𝑓(𝑥𝑛) ∆𝑥 is called the Riemann Sum.

Definite Integral and Riemann Sum

The definite integral of f on the interval a≤x≤b, denoted by 𝑎

𝑏𝑓 𝑥 𝑑𝑥 is the limit of Riemann Sum as

𝑛 → ∞, that is

න𝑎

𝑏

𝑓 𝑥 𝑑𝑥 = lim𝑛→∞

𝑓(𝑥1) + 𝑓(𝑥2) + ⋯+ 𝑓(𝑥𝑛) ∆𝑥

The function f(x) is called the integrand, and the numbers a and b are called the lower and upper limits of integration, respectively. The process of finding a definite integral is called definite integration.

Area as a Definite Integral

If f(x) is continuous and f(x) ≥ 0 on the interval a≤x≤b, thenthe region R under the curve y = f(x) over the interval a≤x≤bhas area A given by the definite integral

𝐴 = න𝑎

𝑏

𝑓 𝑥 𝑑𝑥

The Fundamental Theoremof Calculus

If the function f(x) is continuous on the interval a ≤ x

≤ b, then

𝑎𝑏𝑓 𝑥 𝑑𝑥 = 𝐹 𝑏 − 𝐹(𝑎),

where F(x) is any antiderivative of f(x) on a ≤ x ≤ b.

Another notation:

𝑎𝑏𝑓 𝑥 𝑑𝑥 = ȁ𝐹 𝑥 𝑏

𝑎= 𝐹 𝑏 − 𝐹(𝑎)

Examples

1. Evaluate 01𝑒−𝑥 + 𝑥 𝑑𝑥.

2. Find the area under the graph of 𝑓 𝑥 =1

5𝑥+1between 𝑥 = 0 and 𝑥 = 3.

Integration Rules

Examples

1. .

2. .

Substituting in a Definite Integral

Example. 02 𝑥2

𝑥3+1𝑑𝑥.

Two Ways of Substituting

Examples

1. .

2. .

5.4 Area Between Curves and Average Value

Area Between Curves

Area Between Curves

Examples

If f(x) and g(x) are continuous with f(x) ≥ g(x) on the interval a ≤ x ≤ b, then the area A between the curves y = f(x) and y = g(x) over the interval is given by

1. Find the area of the region R enclosed by the curves y = x3 and y = x2

2. Find the area of the region bounded by the graph of y = x2 and y = x+2

3. Find the area of the region enclosed by the line y = 4x and y = x3 + 3x2

Lorentz CurvesA Lorentz curve is a device used by both economists and sociologists to measure the percentage of a society’s wealth that is possessed by a given percentage of its people.

The Lorentz curve for a particular society’s economy is the graph of the function 𝐿(𝑥), which denotes the fraction of total annual national income earned by the lowest-paid 100𝑥% of the wage-earners in the society, for 0 ≤ 𝑥 ≤ 1.

Example.

If the lowest-paid 30% of all wage-earners receive 23% of the society’s total income, then 𝐿 0.3 = 0.23.

Properties.

1. 𝐿(𝑥) is an increasing function on the interval 0 ≤ 𝑥 ≤ 1,

2. 0 ≤ 𝐿(𝑥) ≤ 1 because 𝐿(𝑥) is a percentage,

3. 𝐿 0 = 0 because no wages are earned when no wage-earners are employed,

4. 𝐿 0 = 0 because 100% of wages are earned by 100% of the wage-earners,

5. 𝐿(𝑥) ≤ 𝑥 because the lowest-paid 100𝑥% of wage-earners cannot receive more than 100𝑥% of total income.

Lorentz Curve and Gini Index

Complete Equality

The line 𝑦 = 𝑥 represents the ideal case corresponding to complete equality in the distribution of income (wage-earners with the lowest 100𝑥% of income receive 100𝑥% of the society’s wealth).

The closer a particular Lorentz curve is to this line, the more equitable the distribution of wealth in the corresponding society.

The total deviation of the actual distribution of wealth in the society from complete equality is the area of the region 𝑅1 between the Lorentz curve 𝑦 = 𝐿(𝑥) and the line 𝑦 = 𝑥.

Gini Index

The ratio of this area to the area of the region 𝑅2 under the complete equality line 𝑦 = 𝑥 over 0 ≤ 𝑥 ≤ 1 is used as a measure of the inequality in the distribution of wealth in the society.

This ratio is called the Gini index, denoted by 𝐺𝐼 (also called the index of income inequality) is

𝐺𝐼 =𝑎𝑟𝑒𝑎 𝑜𝑓 𝑅1𝑎𝑟𝑒𝑎 𝑜𝑓 𝑅1

= 2න0

1

𝑥 − 𝐿(𝑥) 𝑑𝑥

The Gini index always lies between 0 and 1.

An index of 0 corresponds to total equity in the distribution of income, while an index of 1 corresponds to total inequity (all income belongs to 0% of the population).

The smaller the index, the more equitable the distribution of income, and the larger the index, the more the wealth is concentrated in only a few hands.

Example

The Average Value of a Function

Let f(x) be a function that is continuous on the interval a ≤ x ≤ b. Then

the average value V of f(x) over a ≤ x ≤ b is given by the definite

integral

Example.

Geometric Interpretation

The average value V of f(x) over an interval a ≤ x ≤ b where f(x) is continuous and satisfies f(x)≥0 is equal to the height of a rectangle whose base is the interval and whose area is the same as the area under the curve y = f(x) over a ≤ x ≤ b.

The rectangle with base a ≤ x ≤ band height V has the same area as the region under the curve y = f(x)over a ≤ x ≤ b .

5.5 Additional Applications to Business and Economics

Future Value and Present Value of an Income FlowTerm: A specified time period 0≤t≤T.

An income flow (stream): A stream of income transferred continuously into an account.

Future value of the income stream: Total amount (money transferred into the account plus interest) that is accumulated during the specified term

Annuity: A sequence of discrete deposits that is used to approximate the continuous income stream.

Example

Money is transferred continuously into an account at the constant rate of $1200 per year. The account earns interest at the annual rate of 8% compounded continuously. How much will be in the account at the end of 2 years?

Recall

Step 1. P dollars invested at 8% compounded continuously will be worth Pe0.08t dollars t years later.

Step 2. Divide the 2-year time interval 0 ≤ t ≤ 2 into n equal subintervals of length ∆t years and let tj denote the beginning of the j-th subinterval. Then, during the j-thsubinterval

money deposited = (dollars per year) (number of years) = 1200∆t

Step 3. If all of this money were deposited at the beginning of the subinterval, it would remain in the account for 2 - tjyears and therefore would grow to 1200∆t e0.08(2 – tj) dollars.

Thus,

future value of money deposited during the j-th subinterval 1200∆t e0.08(2 – tj)

Step 4. The future value of the entire income stream is the sum of the future values of the money deposited during each of the n subintervals. Hence

future value of income stream σ𝑗=1𝑛 1,200𝑒0.08(2−𝑡𝑗)∆𝑡.

Step 5. As n increases without bound, the length of each subinterval approaches zero and the approximation approaches the true future value of the income stream. Hence

future value of income stream = lim𝑛→∞

σ𝑗=1𝑛 1,200𝑒0.08(2−𝑡𝑗)∆𝑡

= 021,200𝑒0.08(2−𝑡)𝑑𝑡 =1,200𝑒0.16 0

2𝑒−0.08𝑡𝑑𝑡

= ቚ1,200

−0.08𝑒0.16𝑒−0.08𝑡 2

0= −15,000𝑒0.16𝑒−0.16 + 15,000𝑒0.16

≈ 2,602.66

Future Value of an Income Stream

Suppose money is being transferred continuously into an account over a time period 0 ≤ t ≤ T at a rate given by the function f(t) and that the account earns interest at an annual rate r compounded continuously. Then the future value of the income stream (FV) over the term T is given by the definite integral

Present Value of an Income Stream

The amount of money (A) that must be deposited now at the prevailing interest rate r to generate the same income as the income stream, generated at a continuous rate f(x), over the same T years period.

Since A dollars is invested at annual interest rate r compounded continuously will be worth AerT dollars in T years.....

Present Value of an Income Stream

The present value of an income steam (PV) that is deposited continuously at the rate f(t) into an account that earns interest at an annual rate rcompounded continuously for a term of T years is given by

An Example

Jane is trying to decide between two investments. The first costs $1000 and is expected to generate a continuous income stream at the rate of f1(t) = 3000e0.03t dollars per year. The second investment costs $4000 and is estimated to generate income at the constant rate of f2(t) = 4000dollars per year. If the prevailing annual interest rate remains fixed at 5% compounded continuously over the next 5 years, which investment will generate more net income over this time period?

Hint: The net value of each investment over 5 years period is the present value of the investment less its initial cost. For each investment, we have r = 0.05 and T = 5.

Another Example

Joyce is considering a 5-year investment, and estimates that t years from now it will be generating a continuous income stream of 3,000 + 50t dollars per year.

If the prevailing annual interest rate remains fixed at 4% compounded continuously during the entire 5-year term, what should the investment be worth in 5 years?

Consumer Willingness to Spend

The consumer demand function p = D(q) gives the price p that must be charged for each unit of the commodity if q units are to be sold (demanded).

If A(q) is the total amount that consumers are willing to pay for q units, then the demand function can also be thought of as the rate of change of A(q) with respect to q; that is, A’(q) = D(q).

Integrating and assuming that A(0) = 0 (consumers are willing to pay nothing for 0 units), we find that A(q0), the amount that consumers are willing to pay for q0 units of the commodity, is given by

𝐴 𝑞0 = 𝐴 𝑞0 − 𝐴 0 = 0𝑞0 𝑑𝐴

𝑑𝑞𝑑𝑞 0=

𝑞0𝐷(𝑞)𝑑𝑞.

A(q) is the total willingness to spend and D(q) = A’(q) is the marginal willingness to spend.

Graph of Consumer Willingness to Spend

An Example

Suppose that the consumers’ demand function for a certain commodity is D(q) = 4(25 - q2) dollars per unit.

Find the total amount of money consumers are willing to spend to get 3 units of the commodity.

Consumers’ and Producers’ Surplus

In a competitive economy, the total amount that consumers actually spend on a commodity is usually less than the total amount they would have been willing to spend.

Suppose the market price of a particular commodity has been fixed at p0 and consumers will buy q0 units at that price.

p0 = D(q0),

where D(q) is the demand function for the commodity.

The difference between the consumers’ willingness to pay for q0units and the amount they actually pay, p0q0, is called consumers’ surplus.

.

.

Consumers’ Surplus

Producers’ Surplus

Recall that the supply function p = S(q) gives the price per unit that producers are willing to accept in order to supply qunits to the marketplace. Any producer who is willing to accept less than p0 = S(q0)dollars for q0 units gains from the fact that the price is p0. Then producers’ surplus is the difference between what producers would be willing to accept for supplying q0 units and the price they actually receive.....

An Example

A tire manufacturer estimates that q (thousand) radial tires will be purchased (demanded) by wholesalers when the price is

p = D(q) = 0.1q2 + 90

dollars per tire, and the same number of tires will be supplied when the price is

p = S(q) = 0.2q2 + q + 50

dollars per tire.a. Find the equilibrium price (where supply equals demand) and the quantity supplied and demanded at that price.b. Determine the consumers’ and producers’ surplus at the equilibrium price.