The average value of $f ( x ) = \cos x$ on $\left[ 0 , \frac { \pi } { 6 } \right]$ is

A) $\frac { 2 } { \pi }$
B) $\frac { 3 } { \pi }$
C) $\frac { 4 } { \pi }$
D) $\frac { 5 } { \pi }$
E) $\frac { 6 } { \pi }$

Question

The average value of $f ( x ) = \cos x$ on $\left[ 0 , \frac { \pi } { 6 } \right]$ is

A) $\frac { 2 } { \pi }$
B) $\frac { 3 } { \pi }$
C) $\frac { 4 } { \pi }$
D) $\frac { 5 } { \pi }$
E) $\frac { 6 } { \pi }$

Quizplus · Accepted Answer

The average value of a function $f(x)$ on $[a, b]$ is given by $\frac{1}{b-a} \int_{a}^{b} f(x) \, dx$. For $f(x) = \cos x$ on $[0, \frac{\pi}{6}]$, this becomes $\frac{1}{\frac{\pi}{6} - 0} \int_{0}^{\frac{\pi}{6}} \cos x \, dx = \frac{6}{\pi} [\sin x]_{0}^{\frac{\pi}{6}} = \frac{6}{\pi} (\sin \frac{\pi}{6} - \sin 0) = \frac{6}{\pi} \times \frac{1}{2} = \frac{3}{\pi}$.

The Average Value Of f(x)=cos⁡xf ( x ) = \cos xf(x)=cosx

The Average Value Of $f ( x ) = \cos x$