The average value of $f ( x ) = 1 + \sin x$ on $[ 0 , \pi ]$ is

A) $\frac { 2 + \pi } { 2 }$
B) $\frac { 2 - \pi } { 2 }$
C) $\frac { \pi - 2 } { 2 }$
D) $\frac { 2 + \pi } { \pi }$
E) $\frac { 2 - \pi } { \pi }$

Question

The average value of $f ( x ) = 1 + \sin x$ on $[ 0 , \pi ]$ is

A) $\frac { 2 + \pi } { 2 }$
B) $\frac { 2 - \pi } { 2 }$
C) $\frac { \pi - 2 } { 2 }$
D) $\frac { 2 + \pi } { \pi }$
E) $\frac { 2 - \pi } { \pi }$

Quizplus · Accepted Answer

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

The Average Value Of f(x)=1+sin⁡xf ( x ) = 1 + \sin xf(x)=1+sinx

The Average Value Of $f ( x ) = 1 + \sin x$