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Answer:

Step-by-step explanation:

1.To find the average value of the function f(x) = 4 - x^2 over the interval [-2,2],

 we need to evaluate the definite integral of the function over the interval, and then divide by the length of the interval.

The definite integral of the function over the interval [-2,2] is given by:

∫(from -2 to 2) (4 - x^2) dx = [4x - (x^3)/3] (from -2 to 2) = [32/3]

The length of the interval is 2 - (-2) = 4.

Therefore, the average value of the function over the interval [-2,2] is:

[32/3] / 4 = 8/3

So the answer is: 8/3

2)

The definite integral of the function over the interval [1,3] is given by:

∫(from 1 to 3) 4(x^2+1)/x^2 dx

= ∫(from 1 to 3) 4 + 4/x^2 dx

= [4x + 4(-1)/x] (from 1 to 3)

= 8 + 4/3

= 28/3

The length of the interval is 3 - 1 = 2.

Therefore, the average value of the function over the interval [1,3] is:

(28/3) / 2 = 14/3

So the answer is: 14/3

3)The definite integral of the function over the interval [0,pi] is given by:

∫(from 0 to pi) sin(x) dx = [-cos(x)] (from 0 to pi) = 2

The length of the interval is pi - 0 = pi.

Therefore, the average value of the function over the interval [0,pi] is:

2 / pi

So the answer is: 2/pi

4) The definite integral of the function over the interval [0,pi/2] is given by:

∫(from 0 to pi/2) cos(x) dx = [sin(x)] (from 0 to pi/2) = 1

The length of the interval is pi/2 - 0 = pi/2.

Therefore, the average value of the function over the interval [0,pi/2] is:

1 / (pi/2)

Simplifying, we get:

2/pi

So the answer is: 2/pi

5)The definite integral of the function over the interval [1,3] is given by:

∫(from 1 to 3) 9/x^2 dx = [-9/x] (from 1 to 3) = 3 - (-9) = 12

The length of the interval is 3 - 1 = 2.

Therefore, the average value of the function over the interval [1,3] is:

12 / 2 = 6

So the answer is: 6

6) The definite integral of the function over the interval [1,4] is given by:

∫(from 1 to 4) x^2 dx = [x^3/3] (from 1 to 4) = (4^3/3) - (1^3/3) = 21

The length of the interval is 4 - 1 = 3.

Therefore, the average value of the function over the interval [1,4] is:

21 / 3 = 7

So the answer is: 7