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.
$f (x) = 2 {(1 - x)}^{3 / 2} + 3$ , $[a, b] = [- 2, 0]$

The arc length is $\begin{array}{rcl} \frac{2}{27} {(28)}^{3 / 2} - \frac{2}{27} {(10)}^{3 / 2} \end{array}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

Consider the given function $f (x) = 2 {(1 - x)}^{3 / 2} + 3$ , $[a, b] = [- 2, 0]$ .

Step 2. Use arc length formula.

The formula for a function to find the arc length from $x = a$ to $x = b$ is given by $\int_{a}^{b} \sqrt{1 + (f^{'} {(x))}^{2}} d x$ .

Step 3. Find the arc length.

$\begin{array}{rcl} Arc length & = & \int_{- 2}^{0} \sqrt{1 + {(\frac{d}{dx} (2 {(1 - x)}^{3 / 2} + 3))}^{2}} dx \\ = & \int_{- 2}^{0} \sqrt{1 + {((2 {(\frac{3}{2}) (1 - x)}^{1 / 2} (- 1) + 0))}^{2}} dx \\ = & \int_{- 2}^{0} \sqrt{1 + {(- 3 {(1 - x)}^{1 / 2}))}^{2}} dx \\ = & \int_{- 2}^{0} \sqrt{1 + 9 (1 - x)} dx \\ = & \int_{- 2}^{0} \sqrt{10 - 9 x} dx \end{array}$

Step 4. Simply obtained integral by simple substitution.

Substitute $10 - 9 x = u$ , $d x = - \frac{1}{9} d u$ into $\begin{array}{rcl} \int_{- 2}^{0} \sqrt{10 - 9 x} dx \end{array}$ .

$\begin{array}{rcl} \int_{28}^{10} - \frac{\sqrt{u}}{9} d u & = & (- \frac{1}{9}) \int_{28}^{10} \sqrt{u} d u \\ = & - (- \frac{1}{9}) \int_{10}^{28} \sqrt{u} d u \\ = & (\frac{1}{9}) \int_{10}^{28} (\frac{2}{3} u^{3 / 2}) \\ = & (\frac{1}{9}) {[\frac{2}{3} u^{3 / 2}]}_{10}^{28} \\ = & \frac{2}{27} {(28)}^{3 / 2} - \frac{2}{27} {(10)}^{3 / 2} \end{array}$

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Calculus

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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.
$f (x) = 2 {(1 - x)}^{3 / 2} + 3$ , $[a, b] = [- 2, 0]$

Step by Step Solution

Step 1. Given information.

Step 2. Use arc length formula.

Step 3. Find the arc length.

Step 4. Simply obtained integral by simple substitution.

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Calculus

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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)=2(1-x)3/2+3, a,b=-2,0

Step by Step Solution

Step 1. Given information.

Step 2. Use arc length formula.

Step 3. Find the arc length.

Step 4. Simply obtained integral by simple substitution.

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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.
$f (x) = 2 {(1 - x)}^{3 / 2} + 3$ , $[a, b] = [- 2, 0]$