The average value of $f ( x ) = \sec ^ { 2 } x$ on $\left[ 0 , \frac { \pi } { 4 } \right]$ 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 $f ( x ) = \sec ^ { 2 } x$ on $\left[ 0 , \frac { \pi } { 4 } \right]$ 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 the interval $[a, b]$ is given by $\frac{1}{b-a} \int_{a}^{b} f(x) dx$. For $f(x) = \sec^2x$ on $\left[0, \frac{\pi}{4}\right]$, the integral is $\int_{0}^{\frac{\pi}{4}} \sec^2x dx = \left. \tan x \right|_{0}^{\frac{\pi}{4}} = 1$. The length of the interval is $\frac{\pi}{4} - 0 = \frac{\pi}{4}$, so the average value is $\frac{1}{\frac{\pi}{4}} \times 1 = \frac{4}{\pi}$.

The Average Value Of f(x)=sec⁡2xf ( x ) = \sec ^ { 2 } xf(x)=sec2x

The Average Value Of $f ( x ) = \sec ^ { 2 } x$