Find the average value of the function $f ( x )$ in the interval $[ - \pi , \pi ]$.
$$f ( x ) = \sin ^ { 6 } x \cos ^ { 5 } x$$
A) $\pi$
B) 0
C) $\frac { \pi } { 5 }$
D) $\frac { \pi } { 12 }$
E) $\frac { \pi } { 6 }$

Question

Quizplus · Accepted Answer

Since $-1\leq \sin x,\cos x\leq 1$, we have $$0\leq\sin^6 x\cos^5 x\leq 1.$$
Thus, the average value of the function over the interval $[- \pi , \pi ]$ is between 0 and 1. However, since the function is odd, any positive contributions to the average value on the interval $[-\pi,0)$ are cancelled out by the negative contributions on the interval $(0,\pi]$, so the average value must be zero. Thus, the answer is $\boxed{\textbf{(B)}}$.

Find the Average Value of the Function f(x)f ( x )f(x)

Find the Average Value of the Function $f ( x )$