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How do you find the average value of the function #f(x)= -x sin(4x)# on the integral from 0 to #pi/6#?

Answer 1

Charles Anctil

Have a look (but check my maths!):

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Answer 2

Evelyn Agard

To find the average value of the function (f(x) = -x \sin(4x)) over the interval ([0, \frac{\pi}{6}]), you need to calculate the definite integral of the function over that interval and then divide by the width of the interval.

The average value (A) is given by:

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

where (a) and (b) are the limits of integration.

In this case:

[ A = \frac{1}{\frac{\pi}{6} - 0} \int_{0}^{\frac{\pi}{6}} (-x \sin(4x)) , dx ]

You would then evaluate this integral using integration techniques. Once you find the value of the integral, you divide it by the width of the interval, which is (\frac{\pi}{6}). This gives you the average value of the function over the interval ([0, \frac{\pi}{6}]).

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Answer from HIX Tutor

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.