expression’s derivative is 2x. as such, there are an infinite...

Hartfield MATH 2040 | Unit 6

§5.1a Antiderivatives and Indefinite Integrals

As differentiation is a process, it should be possible to study reversing the process. A formal name for this reverse process is antidifferentiation. Another name for the process of antidifferentiation is indefinite integration. Definite integration is a related concept but its origin as a concept was quite different and will be discussed at a later point. For the expression 2x, an antiderivative of 2x is x2 because the derivative of x2 is 2x.

x2 however is not the only the only antiderivative of 2x. x2 + 3, x2 – 5, or x2 + 12000 are all antiderivatives of 2x since each expression’s derivative is 2x. As such, there are an infinite number of antiderivatives to any given function, differing only by a constant. In general, the letter C is used to represent the unknown constant for an antiderivative.


Indefinite Integrals

An indefinite integral is the most generic antiderivative of an expression. For example,

22 x d x x C

shows that the indefinite integral of 2x is x² + C. ∫ is the integral symbol, 2x is the integrand of the integral, and dx tells us that integration is occurring with respect to x. To find indefinite integrals, we will need to follow some basic rules that are related to differentiation.

Basic Integration Rules

Rule 1: Power Rule for Integration

For any constant exponent n except n = -1,

11

1

n nx d x x C

n

.

Rule 2: Constant Rule for Integration

For any constant k,

k d x k x C .


Ex.: Find each indefinite integral below.

A. 4x d x

B. 3x d x

Ex.: Find each integral below.

C. 2

1d x

x

D. 5 d x


Rule 3: Constant Multiple Rule for Integration For any constant c,

k f x d x k f x d x .

Rule 4a: Sum Rule for Integration

f x g x d x f x d x g x d x

Rule 4b: Difference Rule for Integration

f x g x d x f x d x g x d x


A. 7 3

1 6 2 4x x x d x

B. 3

21 21 2 2 3x x d x



A. 2

2

99 x d x

x

B. 62

3 3 2

6x d x

x


C. 2

3x x d x

D. 5

4 2x xd x

x


§5.2a Indefinite Integrals using Logarithmic & Exponential Functions

Rule 6: Integration of Exponential Functions

For any coefficient a ≠ 0,

1a x a x

e d x e Ca

.

Rule 7: Integration resulting in a Logarithmic Function

11ln

d xd x x d x x C

x x

.

Things to note:

1. Special case of rule 6: .x x

e d x e C

2. Rule 6 does not work if there is e is being

raised to a power of x.

3. Rule 7 closes the hole in rule 1 (n = –1).

4. We will not use a rule that integrates a natural logarithm. For reference,

ln ln .x d x x x x C



A. 4 xe d x

B. / 31 2

xe d x


C. 5

d xx

D. 4

d x

x


Ex.: Find the indefinite integral below.

E. 1 2

1x

e x x d x

Ex.: Find the indefinite integral below.

F.

21x

d xx


§5.1b-5.2b Applications of Indefinite Integration

Since indefinite integration allows us to “un-“differentiate, any functions that can be attributed to differentiation can be reversed by indefinite integration. Ex. 1: A company’s marginal cost function is

4 13 3( ) 2 1 6 5 0 ,M C x x x where x is

the number of units and the fixed costs are $4000.

A: Find the total costs function.

B: Compare the marginal cost when

x = 1000 to the total cost when x = 1000.


Ex. 2: A patient’s temperature is 104 Fahrenheit and is changing at a rate of t² – 3t degrees per hour, where t is the number of hours since taking a fever-reducing medication (0 < t < 3).

A: Find a formula for the patient’s

temperature after t hours.

B: Determine the patient’s

temperature after 3 hours.


Ex. 3: A company installs a new computer

that is expected to generate savings at a rate of 20000e–0.02t dollars per year, where t is the number of years the computer has been in operation.

(Source: 4th edition p. 334, #46)

A: Find a formula for the total savings

that the computer will generate during its first t years.

B: If the computer originally cost $250,000, when will it “pay for itself”?


Ex. 4: A biotechnology investment, originally

worth $20,000, grows continuously at the rate of 1000e0.10t dollars per year, where y is the number of years since the investment was made .


A: Find a formula for the value of the

investment after t years.

B: Use the formula to find the value of the investment after 7 years.


Ex. 5: In an effort to reduce its inventory, a

bookstore runs a sale on its least popular mathematics books. The sales rate (books sold per day) on day t of the sale is predicted to be 60/t (for t > 1), where t = 1 corresponds to the beginning of the sale, at which time none of the inventory of 350 books had been sold.


A: Find a formula for the number of

books sold up to day t.

B: Will the store have sold its inventory of 350 books by day t = 30?


§5.3 Definite Integrals and Area An important question that calculus allows us to answer is how much area is between a curve and the x-axis from one x-value to a second x-value. Consider the curve shown below.

If we were to set boundaries at x = 1 and x = 7, we would define the space seen as follows,

and we could identify the space between the x-axis and the curve from x = 1 to x = 7 as the area shaded below.


Not knowing the equation that creates the curve, it is difficult to know how to find this area. One way to make an estimation is to create rectangles based on points of the function.

I’ve created two rectangles where I’ve measured the height based on the value of the function at the left side of the rectangle. (These are called left-hand rectangles.) The first rectangle overestimates the area while the second one underestimates the area. How much the errors will be is unclear. Maybe using more rectangles would be helpful?


On this next drawing I’ve upped the number of rectangles to six. While there is some error still with each (sometimes overestimating and sometimes underestimating), the amount of error is less. In stands to reason that if I continue to use additional rectangles, I should get an increasingly accurate estimation of the area under the curve.

This process of using rectangles to estimate the area under a curve is an example of “Riemann Sums”. As the number of rectangles grows infinitely large, the difference between the actual area under the curve and the area in the rectangles gets closer to zero. We will not find areas under curves using Riemann Sums because there is a more sophisticated way of finding the area. However a good illustration of understanding Riemann Sums can be found at the following link: http://www.geogebra.org/en/examples/integral/loweruppersum.html

http://www.geogebra.org/en/examples/integral/loweruppersum.html

http://www.geogebra.org/en/examples/integral/loweruppersum.html


Definite Integrals Defintion: The definite integral of a

nonnegative function f from

a to b, represented by ( ) ,b

af x d x

is the area under the curve from a to b. a and b are called the lower and upper limits of integration, respectively.

The definite integral is derived

from taking the limit of the areas found by an increasing number of rectangles under the function with increasingly smaller widths.

The Fundamental Theorem of Calculus Theorem: For any continuous function f on

an interval [a,b],

( ) ( ) ( ) ( ) ,b b

aaf x d x F x F b F a

where F is any antiderivative of f. In practice, finding the value of the definite integral comes from finding the antiderivative of f (and ignoring the +C) and then finding the difference between F at b and F at a.


Ex. 1: Find the definite integrals.

A. 4

4

1x d x

B. 2

2

04

xe d x

Ex. 2: Find the definite integral.

3

2

13 2x x d x


Ex. 1: Find the area under 1

( )f xx

from x =1

to x = 2.

Ex. 2: Find the area under 2( ) 2 7 3f x x

from x =1 to x = 3.


Applications of Definite Integrals A definite integral allows you to calculate a total value function when given a rate. For example, given a marginal cost function MC(x), the total cost of units a to b is equal to

( ) .b

aM C x d x

Ex. 1: An average child of age x years gains

weight at the rate of 3.9x1/2 pounds per year from birth to age 16. Find the total weight gain from age 1 to age 9.



Ex. 2: The marginal cost function for the

manufacture of portable CD players is given by the function below where x is the number of CD players manufactured.

( ) 2 05 0 0 0

xM C x

(Source: Finite Mathematics & Calculus Applied to the Real World (1996) p. 988, #25)

A: How much will it cost to produce CD

players 11-100?

B: Part a found the cost for making 90

CD players. Without calculating the exact cost, should making the next 90 CD players cost more, less, or the same as making CD players 11-100?


Let’s say a curve goes both above and below the x-axis between a and b. Will the definite integral from a to b find the area between the curve and the x-axis?

Ex.: Find the area between the curve and the

x-axis between a and b below.

3( ) 4f x x x over [-1, 2]


§5.4a Area Between Curves To find the area between a pair of curves, two questions should be asked first:

1. Which curve is on top of the other curve?

2. Do the curves ever intersect? To find the area between functions f and g over the interval [a, b] where f > g for the entire interval, then:

A re a b e tw e e n

( ) ( ) a n d o n [ , ]

b

af x g x d x

f g a b

It is possible for two curves to completely bound an area. In that case, find the intersection points (letting the x-value of the intersection points be m and n) and:

A re a b e tw e e n

( ) ( ) a n d

n

mf x g x d x

f g

(assuming f > g).


Ex. 1: Find the area bounded as given below.

23 , 2 , 0 , 3y x y x x x


1, 2 , 1 , 3

xy e y x x x



2

3 1 2 , 2 1 1y x y x



2 24 , 8 2y x y x


Application Ex.: A company expects profits of 60e0.02t

thousand dollars per month but predicts that if it builds a new and larger factory its profits will be 80e0.04t thousand dollars per month, where t is the number of months from now. Find the extra profits resulting from the factory during the first two years. If the new factory will cost $1,000,000, will this cost be paid off within the first two years?



§5.4b Average Value of a Function and Area Between Curves The average value of a function from a to be can be found by the following formula:

A v e ra g e v a lu e 1

( )o f o n [ , ]

b

af x d x

f a b b a

Ex.: Find the average value of 2( ) 4 3f x x x

over [-2, 2].


Application of Average Value Ex.: A colony of bacteria is growing

continuously at a rate of 12% per hour. At 8 a.m. the colony contained 500 bacteria. Between 8 a.m. and 5 p.m., what is the average size of the colony?


§5.5 Consumers’ Surplus and Producers’ Surplus

Definitions: Consumers’ surplus is the total

savings by consumers for purchasing items at prices less than what they would otherwise be willing to pay.

Producers’ surplus is the total

earnings by producers for selling items at prices higher than what they would otherwise be required to charge.

A demand function for a product gives the price at which exactly x units will be sold. (Usually d(x)).

A supply function for a product

gives the price at which exactly x units will be supplied (Usually s(x)).

Market demand occurs when the

demand function intersects the supply function.


Visual definition of consumers’ surplus: For a given a demand function d and demand level A, there exists a market price B = d(A). The consumers’ surplus is the difference in the demand function and the market price.

Visual definition of producers’ surplus: For a given a supply function s and demand level A, there exists a market price B = s(A). The producers’ surplus is the difference in the demand function and the market price.


Finding Consumers’ Surplus and Producers’ Surplus

0

C o n s u m e rs '( )

S u rp lu s

A

d x B d x

Ex : For the given demand function d(x) and

demand level x, find the consumers’ surplus.

1

2( ) 2 0 0 , 3 0 0d x x x

0

P ro d u c e rs '( )

S u rp lu s

A

B s x d x

Ex : For the given supply function s(x) and

demand level x, find the producers’ surplus.

2

( ) 0 .0 6 , 5 0s x x x


Ex. 1: For the demand function d(x) and

supply function s(x):

a. Find the market demand. b. Find the consumers’ surplus at the

market demand. c. Find the producers’ surplus at the

market demand. ( ) 1 2 0 0 .1 6 , ( ) 0 .0 8d x x s x x


Ex. 2: For the demand function d(x) and

supply function s(x):

a. Find the market demand. b. Find the consumers’ surplus at the

market demand. c. Find the producers’ surplus at the

market demand.

2 2( ) 3 6 0 0 .0 3 , ( ) 0 .0 0 6d x x s x x


Ex. 3: The market for luxury birdhouses has a

demand function of d(x) = 200 – 0.02x² dollars. Based on avian regulations, new companies will enter the market at s(x) = 100 + x dollars. In each case x represents the number of birdhouses built and sold. Determine the market demand and price for luxury birdhouses and find the consumers’ surplus and producers’ surplus.

expression’s derivative is 2x. as such, there are an infinite...

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