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,
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 .
10. , .
12. , .
Use change of variable to , then . So
Also change of variable to ,
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 .
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