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Q. 31

Expert-verified

Calculus

Found in: Page 539

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

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

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

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information .

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

Step 2. Formula used .

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

Arc length of $f (x)$ from $x = a$ to $x = b$ is $f (x) = \int_{a}^{b} \sqrt{1 + {(f' (x))}^{2}} d x$

Step 3. Find the arc length .

$f (x) = 3 x + 1 f' (x) = \frac{d}{d x} [f (x)] = 3$

$f (x) = \int_{a}^{b} \sqrt{1 + {(f' (x))}^{2}} d x = \int_{- 1}^{4} \sqrt{1 + {(3)}^{2}} d x = \int_{- 1}^{4} \sqrt{10} d x = {[\sqrt{10} x]}_{- 1}^{4} = [\sqrt{10} \cdot 4 + \sqrt{10} \cdot 1] = 5 \sqrt{10}$

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