6.5 Average Value of a Function

1-8. Find the average value of the function on the given interval.

1. , .

 


2. , .

 


3. , .

 


4. , .

 

      .

      To calculus this integral, use a change of variable , then . So,

      


5. ,

 .

          

          .


6. , .

 .


8. , .

 .

      Use a change of variable , then . So

      

 

9-12. (a) Find the average value of on the given interval.

      (b) Find such that .

      (c) Sketch the graph of and a rectangle whose area is the same as the area under the graph of .


9. ,

(a)

       

       

       

       .

(b)

(c)

 

10. , .

(a) 

(b)

(c)


12. , .

 .

      Use change of variable to , then . So

      

      Also change of variable to ,

      then .

      .

13. If is continuous and , show that takes on the value at least once on the interval .

 Since is continuous and , there exists a constant in such that

      .


14. Find the numbers such that the average value of on the interval is equal to .

 

          

          

          .

      That is

      .


15. In a certain city the temperature (in ) hours after A.M. is modeled by the function

.

   Find the average temperature during the period from A.M. to P.M.

 


16. If a cup of coffee has temperature in a room where the temperature is then, according to Newton's Law of Cooling, the temperature of the coffee after minutes is . What is the average temperature of the coffee during the first half hour?

 

17. The linear density in a rod long is , where is measured in meters from one end of the rod. Find the average density of the rod.

 Let . Then

      . Change of variable to , then . So

      


19. If denotes the average value of on the interval and , show that