$\text { Find the average value of } f ( x ) = \frac { 5 \left( x ^ { 2 } + 5 \right) } { x ^ { 2 } } \text { on the interval } [ 1,3 ] \text {. }$
A) $\frac { 40 } { 3 }$
B) 35
C) $\frac { 25 } { 3 }$
D) $\frac { 80 } { 3 }$
E) 40

Question

$\text { Find the average value of } f ( x ) = \frac { 5 \left( x ^ { 2 } + 5 \right) } { x ^ { 2 } } \text { on the interval } [ 1,3 ] \text {. }$ 
A) $\frac { 40 } { 3 }$
B) 35
C) $\frac { 25 } { 3 }$
D) $\frac { 80 } { 3 }$
E) 40

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 $f(x) = \frac{5(x^2 + 5)}{x^2}$ on $[1, 3]$, we simplify $f(x)$ to $5 + \frac{25}{x^2}$ and integrate: $\frac{1}{3-1} \int_{1}^{3} (5 + \frac{25}{x^2}) dx = \frac{1}{2} [5x - \frac{25}{x}]_{1}^{3} = \frac{1}{2} [(15 - \frac{25}{3}) - (5 - 25)] = \frac{1}{2} [\frac{40}{3} + 20] = \frac{40}{3}$.

Find the average value of f(x)=5(x2+5)x2 on the interval [1,3]. \text { Find the average value of } f ( x ) = \frac { 5 \left( x ^ { 2 } + 5 \right) } { x ^ { 2 } } \text { on the interval } [ 1,3 ] \text {. } Find the average value of f(x)=x25(x2+5)​ on the interval [1,3].

$\text { Find the average value of } f ( x ) = \frac { 5 \left( x ^ { 2 } + 5 \right) } { x ^ { 2 } } \text { on the interval } [ 1,3 ] \text {. }$