5.4  Indefinite Integrals and the Total Change Theorem

 

Indefinite Integral

         

 

Example

#2  Verify by differentiation

         

 

 

 

 

 

 

 

 

Definite Integrals versus Indefinite Integrals

 

"Family of functions"

 

Find the general antiderivative integral.  Illustrate by graphing several members of the family on the same screen.

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Table of Indefinite Integrals  - page 325

 

Convention - When the formula for general antiderivative is written, it is only valid on intervals where defined.  For example:   if on

 

 

Find the general indefinite integral.

**Sometimes ALGEBRA is necessary first**

#6                                                                    

 

 

 

 

 

 

 

 

 

 

#16  

 

 

 

Evaluating Some Definite Integrals:

#24                                                   

 

 

 

 

 

 

 

 

 

 

#36 

 

 

 

 

 

 

 

 

 

         

 

 

 

 

 

 

 

 

 

 

 

Total Change Theorem

The integral of a rate of change is the total change: 

 

Read through the examples on page 327 on your own.

 

#50  A honeybee population starts with 100 bees and increases at a rate of bees per week.  What does  represent?

 

 

 

 

 

#58  Find a) velocity and b) distance traveled if

; v(0) = -4;  0 < t < 3     

Note:  displacement versus distance traveled .

 

 

 

 

 

 

 

 

 

 

 

 

#60 Water flows from the bottom of a storage tank at a rate of  liters per minute, where .  Find the amount of water that flows from the tank during the first 10 minutes.

 

 

 

 

 

 

 

 

 

 

 

 

Assignment:5.4; pg.329; 1, 5-15 odd, 19-41 odd, 47-51 odd, 55, 57