6.5 Average Value of a Function

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

1. , .

2. , .

3. , .

4. , .

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

5. ,

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6. , .

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8. , .

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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)

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10. , .

(a)

(b)

(c)

12. , .

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Use change of variable to , then . So

Also change of variable to ,

then .

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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

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

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That is

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

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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