Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Consider the region between the graph of $f (x) = x - 2$ and the x-axis on [2,5]. For each line of rotation given in Exercises 35– 40, use definite integrals to find the volume of the resulting solid.

The required volume is $36 π$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information

The given figure is

Step 2. Calculation

Express the curve as inverse function.

$y = x - 2 x = y + 2 g (y) = y + 2$

For the x-interval of $[2, 5]$ , the corresponding interval of y-variable will be $[0, 3]$

For the washer the external radius of each washer is $x = 5$ and internal radius of each wahser is given by $g (y)$

Step 3. Calculation

Determine the volume of solid of revolution.

$V = π \int_{0}^{3} (5^{2} - {(g (y))}^{2}) dx = π \int_{0}^{3} (25 - {(y + 2)}^{2}) dy = π \int_{0}^{3} (25 - y^{2} - 4 y - 4) dy = π \int_{0}^{3} (21 - y^{2} - 4 y) dy = π {[21 y - \frac{y^{3}}{3} - \frac{4 y^{2}}{2}]}_{0}^{3} dy = π ((21 (3) - \frac{3^{3}}{3} - \frac{4 {(3)}^{2}}{2}) - (21 (0) - \frac{0^{3}}{3} - \frac{4 {(0)}^{2}}{2})) = π [(63 - 9 - 18) - 0] = 36 π$

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Calculus

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Short Answer

Consider the region between the graph of $f (x) = x - 2$ and the x-axis on [2,5]. For each line of rotation given in Exercises 35– 40, use definite integrals to find the volume of the resulting solid.

Step by Step Solution

Step 1. Given Information

Step 2. Calculation

Step 3. Calculation

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Short Answer

Consider the region between the graph of f(x)=x-2 and the x-axis on [2,5]. For each line of rotation given in Exercises 35– 40, use definite integrals to find the volume of the resulting solid.

Step by Step Solution

Step 1. Given Information

Step 2. Calculation

Step 3. Calculation

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Consider the region between the graph of $f (x) = x - 2$ and the x-axis on [2,5]. For each line of rotation given in Exercises 35– 40, use definite integrals to find the volume of the resulting solid.