To calculate this integral, use a change of variable , then . So,
Use a change of variable , then . So
8-10. (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 .
Use change of variable to , then . So
Also change of variable to ,
11. 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
12. Find the numbers such that the average value of on the interval is equal to .
That is .
13. 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.
14. 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?
15. 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
16. If denotes the average value of on the interval and ,