Questions: Find the average value of the function (f(x)) in the interval ([-π, π]). (f(x)=sin^3 x cos^3 x)

Find the average value of the function \( f(x) = \sin^3 x \cos^3 x \) over the interval \([- \pi, \pi]\).

Calculate the integral of the function over the interval.

The integral of \( f(x) \) from \( -\pi \) to \( \pi \) is given by: \[ \int_{-\pi}^{\pi} \sin^3 x \cos^3 x \, dx = 0 \] This is because the function \( f(x) \) is an odd function, and the integral of an odd function over a symmetric interval around zero is zero.

Determine the average value of the function.

The average value \( \text{Avg}(f) \) is calculated as: \[ \text{Avg}(f) = \frac{1}{\pi - (-\pi)} \int_{-\pi}^{\pi} f(x) \, dx = \frac{1}{2\pi} \cdot 0 = 0 \]

The average value of the function is \( \boxed{0} \).

The average value of the function \( f(x) = \sin^3 x \cos^3 x \) over the interval \([- \pi, \pi]\) is \( \boxed{0} \).