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.
$f (x) = \sqrt{9 - x^{2}}$ , $[a, b] = [- 3, 3]$

The arc length is $3 π$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

Consider the given function $f (x) = \sqrt{9 - x^{2}}$ , $[a, b] = [- 3, 3]$ .

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 role="math" localid="1649165845561" $\int_{a}^{b} \sqrt{1 + {(f^{'} (x))}^{2}} d x$ .

Step 3. Find the arc length.

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

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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.
$f (x) = \sqrt{9 - x^{2}}$ , $[a, b] = [- 3, 3]$

Step by Step Solution

Step 1. Given information.

Step 2. Use arc length formula.

Step 3. Find the arc length.

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Calculus

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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)=9-x2, a,b=-3,3

Step by Step Solution

Step 1. Given information.

Step 2. Use arc length formula.

Step 3. Find the arc length.

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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.
$f (x) = \sqrt{9 - x^{2}}$ , $[a, b] = [- 3, 3]$