Q. 53 Consider聽the聽region聽between聽... [FREE SOLUTION]

Chapter 6: Q. 53 (page 512)

$Consider the region between the graphs of f (x) = {(x 鈭� 2)}^{2} and g (x) = x on [1, 4] . For each line of rotation given in Exercises 51 鈥� 54, use definite integrals to find the volume of the resulting solid .$

Short Answer

Expert verified

$The volume of solid of revolution, V = \frac{117}{5} 蟺$

Step by step solution

Step 1. Given information is:

$The given region bounded by f (x) = {(x - 2)}^{2} and g (x) = x, between the interval [1, 4], is rotated around the horizontal line, y = - 1 .$

Step 2. Determining Volume of Solid of Revolution

$For the washer, the external radius of each washer is 1 + g (x) and internal radius of each washer is given as 1 + f (x) . The volume of washer is given by : V = 蟺 {鈭�}_{a}^{b} ({(R (x))}^{2} - {(r (x))}^{2}) dx Using this definition to determine the volume of solid of revolution, V = 蟺 {鈭�}_{1}^{4} ({(1 + g (x))}^{2} - {(1 + f (x))}^{2}) dx V = 蟺 {鈭�}_{1}^{4} ({(1 + x)}^{2} - {({1 + (x - 2)}^{2})}^{2}) dx V = 蟺 {鈭�}_{1}^{4} ({(1 + x)}^{2} - {1 - (x - 2)}^{4} - 2 {(x - 2)}^{2}) dx V = 蟺 {[\frac{{(1 + x)}^{3}}{3} - x - \frac{{(x - 2)}^{5}}{5} - \frac{2 {(x - 2)}^{3}}{3}]}_{1}^{5} V = 蟺 [(\frac{125}{3} - 4 - \frac{32}{5} - \frac{16}{3}) - (\frac{8}{3} - 1 + \frac{1}{5} + \frac{2}{3})] V = \frac{117}{5} 蟺$

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91影视

$Consider the region between the graphs of f (x) = {(x 鈭� 2)}^{2} and g (x) = x on [1, 4] . For each line of rotation given in Exercises 51 鈥� 54, use definite integrals to find the volume of the resulting solid .$

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)=(x鈭�2)2andg(x)=xon[1,4].ForeachlineofrotationgiveninExercises51鈥�54,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

Geometry

Applied Mathematics

Decision Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

$Consider the region between the graphs of f (x) = {(x 鈭� 2)}^{2} and g (x) = x on [1, 4] . For each line of rotation given in Exercises 51 鈥� 54, use definite integrals to find the volume of the resulting solid .$