Q. 50 Consider聽the聽region聽between聽... [FREE SOLUTION]

Chapter 6: Q. 50 (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

$The volume of solid of revolution, V = \frac{16}{3} 蟺$

Step by step solution

Step 1. Given information is:

$The given region bounded by f (x) = x^{2} and g (x) = 2 x, between the interval [0, 2], is rotated around vertical line, x = 3 .$

Step 2. Determining Inverse function

$f (x) = x^{2} y = x^{2} x = \sqrt{y} p (y) = \sqrt{y} Determining the inverse of other function also, g (x) = 2 x y = 2 x x = \frac{y}{2} q (y) = \frac{y}{2}$

Step 3. Determing y-interval

$For the x - interval of [0, 2], the corresponding interval of y - variable will be [0, 4]$

Step 4. Determining Volume of Solid of Revolution

$For the washer, the external radius of each washer is 3 - q (y) and internal radius of each washer is given as 3 - p (y) . The volume of washer is given by : V = 蟺 {鈭�}_{a}^{b} ({(R (y))}^{2} - {(r (y))}^{2}) dy Using this definition to determine the volume of solid of revolution, V = 蟺 {鈭�}_{0}^{4} ({(3 - q (y))}^{2} - {(3 - p (y))}^{2}) dy V = 蟺 {鈭�}_{0}^{4} ({(3 - \frac{y}{2})}^{2} - {(3 - \sqrt{y})}^{2}) dy V = 蟺 {鈭�}_{0}^{4} ((9 - 3 y + \frac{y^{2}}{4}) - (9 - 6 \sqrt{y} + y)) dy V = 蟺 {鈭�}_{0}^{4} (- 4 y + \frac{y^{2}}{4} + 6 \sqrt{y}) dy V = 蟺 {[- \frac{4 y^{2}}{2} + \frac{y^{3}}{12} + 6 \frac{y^{\frac{3}{2}}}{\frac{3}{2}}]}_{0}^{4} V = 蟺 [(- 32 + \frac{16}{3} + 32) - (0)] V = \frac{16}{3} 蟺$

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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 Inverse function

Step 3. Determing y-interval

Step 4. 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 Inverse function

Step 3. Determing y-interval

Step 4. Determining Volume of Solid of Revolution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Decision Maths

Geometry

Mechanics Maths

Applied Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

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