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.

$f\left(x\right)=3x+1,\left[a,b\right]=\left[-1,4\right]$

Short answer

The arc length is$5\sqrt{10}$.

Step-by-step solution

Step 1. Given information .

Consider the given function$f\left(x\right)=3x+1$.

Step 2. Formula used .

The formula used to find the arc length of the definite integral is ,

Arc length of $f\left(x\right)$from $x=a$to $x=b$is$f\left(x\right)={\int }_a^b\sqrt{1+{\left(f\text{'}\left(x\right)\right)}^2}dx$

Step 3. Find the arc length .

$$\begin{gathered} f\left(x\right)=3x+1 \\ f\text{'}\left(x\right)=\frac{d}{dx}\left[f\left(x\right)\right]=3 \end{gathered}$$

$$\begin{gathered} f\left(x\right)={\int }_a^b\sqrt{1+{\left(f\text{'}\left(x\right)\right)}^2}dx \\ ={\int }_{-1}^4\sqrt{1+{\left(3\right)}^2}dx \\ ={\int }_{-1}^4\sqrt{10}dx \\ ={\left[\sqrt{10}x\right]}_{-1}^4 \\ =\left[\sqrt{10}·4+\sqrt{10}·1\right] \\ =5\sqrt{10} \end{gathered}$$