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Q. 31

Expert-verified
Found in: Page 539

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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

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

See the 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 .

$f\left(x\right)=3x+1\phantom{\rule{0ex}{0ex}}f\text{'}\left(x\right)=\frac{d}{dx}\left[f\left(x\right)\right]=3$

$f\left(x\right)={\int }_{a}^{b}\sqrt{1+{\left(f\text{'}\left(x\right)\right)}^{2}}dx\phantom{\rule{0ex}{0ex}}={\int }_{-1}^{4}\sqrt{1+{\left(3\right)}^{2}}dx\phantom{\rule{0ex}{0ex}}={\int }_{-1}^{4}\sqrt{10}dx\phantom{\rule{0ex}{0ex}}={\left[\sqrt{10}x\right]}_{-1}^{4}\phantom{\rule{0ex}{0ex}}=\left[\sqrt{10}·4+\sqrt{10}·1\right]\phantom{\rule{0ex}{0ex}}=5\sqrt{10}$