# How do you find the average value of #f(x)=cosx# as x varies between #[0, pi/2]#?

Here, this gives us an average value of:

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To find the average value of ( f(x) = \cos x ) over the interval ( [0, \frac{\pi}{2}] ), you use the formula for the average value of a function over an interval ( [a, b] ), which is:

[ \text{Average value} = \frac{1}{b-a} \int_{a}^{b} f(x) , dx ]

Plugging in the values for this case, where ( a = 0 ) and ( b = \frac{\pi}{2} ), and ( f(x) = \cos x ), we get:

[ \text{Average value} = \frac{1}{\frac{\pi}{2} - 0} \int_{0}^{\frac{\pi}{2}} \cos x , dx ]

[ \text{Average value} = \frac{2}{\pi} \int_{0}^{\frac{\pi}{2}} \cos x , dx ]

[ \text{Average value} = \frac{2}{\pi} [\sin x]_{0}^{\frac{\pi}{2}} ]

[ \text{Average value} = \frac{2}{\pi} [\sin \frac{\pi}{2} - \sin 0] ]

[ \text{Average value} = \frac{2}{\pi} [1 - 0] ]

[ \text{Average value} = \frac{2}{\pi} ]

So, the average value of ( f(x) = \cos x ) over the interval ( [0, \frac{\pi}{2}] ) is ( \frac{2}{\pi} ).

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