Q. 49 Consider聽the聽region聽between聽... [FREE SOLUTION]

Chapter 6: Q. 49 (page 512)

$Consider the region between the graphs of f (x) = x^{2} and g (x) = 2 x on [0, 2] . For each line of rotation given in Exercises 47 鈥� 50, use definite integrals to find the volume of the resulting solid .$

Short Answer

Expert verified

$The volume of solid of revolution, V = \frac{104}{15} 蟺$

Step by step solution

Step 1. Given information is:

$The given region bounded by f (x) = x^{2} and g (x) = 2 x, between the interval [0, 2], is rotated around horizontal line, y = - 1 .$

Step 2. Determining Volume of Solid of Revolution

$For the washer, the external radius of each washer is 1 + g (x) and internal radius of each washer is given as 1 + f (x) . The volume of washer is given by : V = 蟺 {鈭�}_{a}^{b} ({(R (x))}^{2} - {(r (x))}^{2}) dx Using this definition to determine the volume of solid of revolution, V = 蟺 {鈭�}_{0}^{2} ({(1 + g (x))}^{2} - {(1 + f (x))}^{2}) dx V = 蟺 {鈭�}_{0}^{2} ({(1 + 2 x)}^{2} - {(1 + x^{2})}^{2}) dx V = 蟺 {鈭�}_{0}^{2} ((1 + 4 x^{2} + 4 x) - (1 + x^{4} + 2 x^{2})) dx V = 蟺 {鈭�}_{0}^{2} (2 x^{2} + 4 x - x^{4}) dx V = 蟺 {[\frac{2 x^{3}}{3} + \frac{4 x^{2}}{2} - \frac{x^{5}}{5}]}_{0}^{2} V = 蟺 [(\frac{16}{3} + 8 - \frac{32}{5}) - (0)] V = \frac{104}{15} 蟺$

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91影视

$Consider the region between the graphs of f (x) = x^{2} and g (x) = 2 x on [0, 2] . For each line of rotation given in Exercises 47 鈥� 50, use definite integrals to find the volume of the resulting solid .$

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Determining Volume of Solid of Revolution

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91影视

Considertheregionbetweenthegraphsoff(x)=x2andg(x)=2xon[0,2].ForeachlineofrotationgiveninExercises47鈥�50,usedefiniteintegralstofindthevolumeoftheresultingsolid.

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Determining Volume of Solid of Revolution

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Applied Mathematics

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Statistics

Geometry

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

$Consider the region between the graphs of f (x) = x^{2} and g (x) = 2 x on [0, 2] . For each line of rotation given in Exercises 47 鈥� 50, use definite integrals to find the volume of the resulting solid .$