# What is the average value of a function #sinx# on the interval #[0, pi/3]#?

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To find the average value of the function ( \sin(x) ) on the interval ([0, \frac{\pi}{3}]), you use the formula:

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

where (a) is the lower limit of the interval (0 in this case) and (b) is the upper limit of the interval (( \frac{\pi}{3} ) in this case).

[ \text{Average value} = \frac{1}{\frac{\pi}{3} - 0} \int_{0}^{\frac{\pi}{3}} \sin(x) , dx ]

[ = \frac{1}{\frac{\pi}{3}} \left[ -\cos(x) \right]_{0}^{\frac{\pi}{3}} ]

[ = \frac{1}{\frac{\pi}{3}} \left( -\cos\left(\frac{\pi}{3}\right) - (-\cos(0)) \right) ]

[ = \frac{1}{\frac{\pi}{3}} \left( -\frac{1}{2} - (-1) \right) ]

[ = \frac{1}{\frac{\pi}{3}} \left( 1 - \frac{2}{2} \right) ]

[ = \frac{1}{\frac{\pi}{3}} \left( \frac{1}{2} \right) ]

[ = \frac{3}{2\pi} ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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