MAT 166 Calculus for Bus/Soc Chapter 7 Notes Antiderivatives - - PowerPoint PPT Presentation

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MAT 166 Calculus for Bus/Soc Chapter 7 Notes Antiderivatives - - PowerPoint PPT Presentation

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6/28/2013 MAT 166 Calculus for Bus/Soc Chapter 7 Notes Antiderivatives Integration David J. Gisch Antiderivative Antiderivative 1 6/28/2013 The Power Rule Antiderivative (Integral) Type equation here. 3 2

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MAT 166 – Calculus for Bus/Soc Chapter 7 Notes

Integration David J. Gisch

Antiderivatives

Antiderivative Antiderivative

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The Power Rule

  • Type equation here.

Antiderivative (Integral)

3 2 10 3 2 10 3 2 10 3 5 2 4 10 1 3 5 1 2 10

Antiderivative

Example: Find

  • .

Antiderivative

Example: Find 6 8 9.

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Antiderivative

Example: Find

  • .

Exponential Integral Integral Antiderivative

Example: Find

  • .
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Integral in Physics

Position

Derivative

Velocity

Derivative

Acceleration Position Velocity Acceleration

Integral Integral To find we need some information. *Note that our book uses rather than .

Integral in Physics

Example: For a particular object, 5 4 and 0 6. Find the equation for velocity.

Integral in Physics

Example: Suppose 9 3 and 1 8. Find the equation for speed.

Substitution

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S ubstitution

  • The derivative has several rules but it is fairly

straightforward, relatively speaking.

  • Going in reverse, finding the integral (antiderivative), is
  • ften less straightforward.

Antiderivative

Example: Find 2 1 6. 2 1 6

  • 5

2 1 5

Antiderivative

Example: Find 8 4 8 .

Antiderivative

Example: Find 3 10.

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Antiderivative

Example: Find

  • .

Integral S ubstitution Rules Integral S ubstitution Rules

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Antiderivative

Example: A company incurs debt at a rate of 90 6 12 dollars per year, where t is the amount of time (in years) since the company began. By the fourth year the company had accumulated $16,260 in debt. a) Find the total debt function. b) How many years must pass before the total debt exceeds $40,000?

Debt Example

90 6 12

Area and Definite Integral

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Integral in Physics

Position

Derivative

Velocity

Derivative

Acceleration Position Velocity Acceleration

Integral Integral To find we need some information. *Note that our book uses rather than .

Integral

Total amount of substance leaked Tt Rate of leak,

Integral Integral

Area and the Integral

  • The area under a curve is a visual representation of the

integral.

Area Under a Curve

  • Suppose that we have the following function and it gives

the rate at which water is leaking from a steel drum.

  • The area underneath the curve represents the amount of

leakage.

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Left Hand Rule

  • We could estimate the area by using rectangles whose

heights are determined by the value of the function on the left side.

Right Hand Rule

  • We could estimate the area by using rectangles whose

heights are determined by the value of the function on the right side.

Midpoint Rule

  • We could estimate the area by using rectangles whose

heights are determined by the value of the midpoint.

Area and the Integral

  • Of course we only used two rectangle but if you use more

it is obvious that the estimate for the area becomes more accurate.

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Area and the Integral

  • If you use more rectangles the estimate for the area

becomes more accurate.

The Definite Integral Area and the Integral

1 4 2

Definite Integral

Example: Find the area under the curve from 0 to 4 where 2 and 4, using the midpoint rule.

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Definite Integral

Example: Find the area under the curve from 0 to 6 where 0.5 and 3, using the left hand rule.

Area and Total Change

  • To summarize, if we have a function giving us the rate,

the area under that curve (the integral) gives us the total change.

Definite Integral

Example: The figure below shows the rate oil is leaking from a machine in a large factory (in cubic centimeters per hour) with specific rates over a 12-hour period given in the

  • table. Approximate the total amount of leakage over a 12-

hour shift.

Definite Integral

1 ∙ 15.2 1 ∙ 18 1 ∙ 18.8 1 ∙ 14.1 ⋯ 1 ∙ 16.6 1 ∙ 16.4 187.8 This is an estimation of the definite integral

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The Fundamental Theorem of Calculus

The Fundamental Theorem

  • I prefer to write it as

|

  • 2
  • 2

4 4 0 1 2 4 1 2 0 128

Definite Integral Definite Integral

Example: Find the following definite integral

  • 3

2 2

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Definite Integral

Example: Find the following definite integral

  • 3 3 1
  • 0,15 , 0,450

Definite Integral

Example: Find the following definite integral

  • 1

16 4 8

  • Remember the integral is the area “under” the curve

(from the x-axis ).

  • The area should be positive so we take the absolute value
  • f integrals where the graph is below the x-axis.
  • 4
  • 3 4

2 2 3 4 2 0 3 40 16 3

  • 4
  • 16

3 16 3

Definite Integral

  • 4
  • 4
  • 4
  • 16

3 32 3 16

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Definite Integral Definite Integral

Example: Find the following definite integral

  • Definite Integral
  • Page 397 #69

40.2 3.50 0.897 a) Find the integral of over the interval 0,9. What does this integral represent? b) Baby boomers are those born between 1945 and 1965, that is, those in the range of 4.5 to 6.5 decades in 2010. Estimate the number of baby boomers.

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Page 397 #69

The Area Between Two Curves

Area Between Two Curves

  • We can also find the area between two curves.

Area Between Two Curves

Example: Find the area between the graphs below.

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Area Between Two Curves

Example: Find the area between the graphs below.

Area Between Two Curves

  • The fact that one of the curves is below the x-axis does

not effect the outcome in this scenario.

Area Between Two Curves

  • Where it does matter is when the functions switch from

to or vice versa.

2

  • 2
  • S

avings Analysis

Example: A company is considering a new manufacturing process in one of its plants. The new process provides substantial initial savings, with the savings declining with time (in years) according to the rate-of-savings function 100 , where ′ is in thousands of dollars per year. At the same time, the cost of operating the new process increases with time (in years), according to the rate-of-cost function (in thousands of dollars per year) 14 3 .

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S avings Analysis

a) For how many years will the company realize savings? b) What will the net total savings be during that period?

Consumers’ S urplus

  • Recall that the equilibrium point is where the supply and

demand curves intersect.

▫ Remember that this points represents the price where consumers will purchase the same amount of product that the manufacturer wants to sell.

  • However, some people would be willing to pay more

than the equilibrium price for a product. The difference in these two amounts can be thought of as a “savings” realized by the customer.

▫ This is called the consumers’ surplus.

Consumers’ S urplus

  • Here is the equilibrium price and is the equilibrium

quantity.

  • The integral
  • would be the total amount the

consumers would be willing to pay for items.

Consumers’ S urplus

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Producers’ S urplus Oat Bran

Example: Suppose the price (in dollars per ton) for oat bran is 400

⁄

When the demand for the product is q tons. Also, suppose the function

⁄ 1

gives the price (in dollars per ton) when the supply is q

  • tons. Find the consumers’ surplus and the producers’

surplus.

Oat Bran Oat Bran

  • 400

⁄

199.50

.

  • 199.50

⁄ 1 .

  • Consumers’ Surplus

Producers’ Surplus