Find the average value of cos t for t in the intervals `[0, pi]` , `[0,(pi/2)]` , `[0,(pi/4)]` and `[0, 0.01]` .  

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The graph of the function cos t is not a straight line. To find the average value of cost for t in the interval [a, b] the sum of all values of cost t lying in the interval has to be divided by (b - a). The sum is derived by the definite integral `int_(a)^b cos t dt` .

The average is `(int_(a)^b cos t dt)/(b - a)`

=> `(sin b - sin a)/(b - a)`

For t in the given intervals the corresponding value of the average is:

`[0, pi]: A = (sin pi - sin 0)/pi = 0`

`[0, pi/2]: A = (sin pi/2 - sin 0)/(pi/2) = 2/pi`

`[0, pi/4]: A = (sin (pi/4) - sin 0)/(pi/4) = (2*sqrt 2)/pi`

`[0, 0.01] : A = (sin 0.01 - sin 0)/0.01 ~~ 1`

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