Q. 32 - Calculus

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.

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

Step-by-Step Solution

Answer

The arc length is $3\sqrt{2}$ .

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 of $f\left(x\right)={\int }_a^b\sqrt{1+{\left(f\text{'}\left(x\right)\right)}^2}dx$ .

Step 3. Find the arc length .

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

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