What is the average value of sin(x) on the interval -6leqslant xleqslant -1 ?

Question

Answer 1

To find the average value of \( \sin(x) \) on the interval

, we use the formula for the average value of a continuous function \( f(x) \) over an interval

:

In this case, \( f(x) = \sin(x) \),

, and

. So, we need to compute:

Simplify the denominator:

So the expression becomes:

Next, we need to evaluate the integral \( \int_{-6}^{-1} \sin(x) \, dx \). We know that the integral of \( \sin(x) \) is \( -\cos(x) \):

Applying the limits of integration from

to

:

Since \(\cos(-x) = \cos(x)\), this simplifies to:

Now, substitute this back into the expression for the average value:

Thus, the average value of \( \sin(x) \) on the interval

is:

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