Chapter 6: Q. 48 (page 512)

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

Short Answer

Expert verified

$The volume of solid of revolution, V = \frac{64}{15} π$

Step by step solution

Step 1. Given information is:

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

Step 2. Determining Volume of Solid of Revolution

$For the washer, the external radius of each washer is g (x) and internal radius of each washer is given as f (x) . The volume of washer is given by : V = π \int_{a}^{b} ({(R (x))}^{2} - {(r (x))}^{2}) dx Using this definition to determine the volume of solid of revolution, V = π \int_{0}^{2} ({(g (x))}^{2} - {(f (x))}^{2}) dx V = π \int_{0}^{2} ({(2 x)}^{2} - {(x^{2})}^{2}) dx V = π \int_{0}^{2} (4 x^{2} - x^{4}) dx V = π {[\frac{4 x^{3}}{3} - \frac{x^{5}}{5}]}_{0}^{2} V = π [(\frac{32}{3} - \frac{32}{5}) - (0)] V = \frac{64}{15} π$

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

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Determining Volume of Solid of Revolution

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

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Determining Volume of Solid of Revolution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Statistics

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Study anywhere. Anytime. Across all devices.

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