5.3           The Definite Integral and the Fundamental Theorem of Calculus

 

“When faced with something you don’t know how to handle, try to relate it to something you do know how to handle”

 

The Area of a region under a curve  -  can be viewed as sum of many rectangles –

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Picture Page 388

 

 

The Definite Integral      f(x) a continuous function on [a,b].  Subdivide the interval into n equal parts, each of width .  A number xk from the kth subinterval (for k = 1, 2, 3, ... n)

 

      Riemann Sum      

 

 

The Definite Integral        

 

 

Terminology: 

          Integrand – f(x)              Upper and lower limits – a and b       

 

Area as Definite Integral  -  top of page 399

 

Fundamental Theorem of Calculus

          If the function f(x) is continuous on [a,b], then  

where F(x) is any antiderivative of f(x) on [a,b]

 

 

Examples:

                                                           

 

 

 

 

 

 

RULES – Similar to indefinite integral for constant multiple, sum and difference  -  page 404

 

Additional Rules

4.     

 

5.     

 

6.     

 

Examples:

                                          

 

 

 

 

                                                         


                        

 

 

 

 

 

 

Example     

 If     

      

#46  Find the area of the region R that lies under the given curve y = f(x) over the indicated interval [a,b]

                    Under

 

 

 

 

 

 

Applications

 

#54  MARGINAL COST

The marginal cost of producing a certain commodity is C’(q) = 6q +1 dollars per unit when q units are being produced. 

a)     What is the total cost of producing the first 10 units.

 

 

 

b)     What is the cost of producing the next 10 units?

 

#52  WATER POLLUTION

It is estimated that t years from now the population of a certain lake-side community will be changing at the rate of   thousand people per year.  Environmentalists have found that the level of pollution in the lake increases at the rate of approximately 5 units per 1,000 people.  By how much will the pollution in the lake increase during the next 2 years?

 

 

 

 

 

 

Substitution and Definite Integrals    There are 2 methods

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Application:

The weekly marginal cost of producing x pairs of tennis shoes is given by   where C(x) is cost in dollars.

a) If the fixed costs are $2,000 per week, find the cost function.

 

 

 

 

b)  What is the average cost per pair of shoes if 1,000 pairs of shoes are produced each week?

 

 

#60  An advertising agency begins a campaign to promote a new product and determines that t days later, the number of people N(t) who have heard about the product is changing at a rate given by  people per day.  How many people learn about the product during the first week?  During the second week?

 

 

 

 

Assignment:  5.3, page 410, 1-23 odd, 27-47 odd, 51, 53, 55