# What is the average value of a function #y=6/x# on the interval #[1,e]#?

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To find the average value of the function ( y = \frac{6}{x} ) on the interval ([1,e]), we need to evaluate the definite integral of the function over that interval and then divide by the length of the interval.

The integral of ( y = \frac{6}{x} ) with respect to ( x ) is ( 6\ln(x) ).

So, the average value ( \bar{y} ) of the function on the interval ([1,e]) is:

[ \bar{y} = \frac{1}{e-1} \int_{1}^{e} \frac{6}{x} , dx = \frac{1}{e-1} \left[ 6\ln(x) \right]_{1}^{e} ]

[ = \frac{1}{e-1} \left[ 6\ln(e) - 6\ln(1) \right] ]

[ = \frac{1}{e-1} \left[ 6 - 0 \right] ]

[ = \frac{6}{e-1} ]

So, the average value of the function ( y = \frac{6}{x} ) on the interval ([1,e]) is ( \frac{6}{e-1} ).

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