Question

Find the exact value of the arc length of each function f (x) on [a, b] by writing the arc length as a definite integral and then solving that integral.

$f\left(x\right)=4-x$ ,$\left[a,b\right]=\left[2,5\right]$

Short answer

The arc length is $3\sqrt{2}$.

Step-by-step solution

Step 1. Given information .

Consider the given function$f\left(x\right)=4-x$.

Step 2. Formula used .

The formula used to find the arc length from $x=a$to $x=b$is ,

Arc length oflocalid="1649150288054" $f\left(x\right)={\int }_a^b\sqrt{1+{\left(f\text{'}\left(x\right)\right)}^2}dx$.

Step 3. Find the arc length .

$$\begin{gathered} f\left(x\right)=4-x \\ f\text{'}\left(x\right)=\frac{d}{dx}\left[f\left(x\right)\right]=-1 \end{gathered}$$

$$\begin{gathered} Arclength={\int }_2^5\sqrt{1+{\left(-1\right)}^2}dx \\ ={\left[\sqrt{2}x\right]}_2^5 \\ =\left[\sqrt{2}·5-\sqrt{2}·2\right] \\ =5\sqrt{2}-2\sqrt{2} \\ =3\sqrt{2} \end{gathered}$$