Q. 30 Consider the region between f(x)... [FREE SOLUTION]

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Chapter 6: Q. 30 (page 510)

Consider the region between $f (x) = \sqrt{x}$ and the x-axis on $[0, 4]$ . For each line of rotation given in Exercises 27鈥�30, use four disks or washers based on the given rectangles to approximate the volume of the resulting solid.

Short Answer

Expert verified

The required volume is $\frac{583}{16} 蟺$

Step by step solution

Step 1. Given Information

The given figure is

Step 2. Calculation

To determine, the volume of solid of revolution rotated around vertical lines express the curve as inverse function.

$f (x) = \sqrt{x} y = \sqrt{x} y^{2} = x x = y^{2} g (y) = y^{2}$

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

The width of each washer is determined as,

$鈭� y = \frac{b - a}{n} = \frac{2 - 0}{4} = \frac{1}{2}$

The external radius is $5 - g (y_{k})$ and inner radius is given as 1.

The starting value of disk is 0. It can be termed as $y_{0}$ .

Determine these end points as follows,

$y_{k} = y_{0} + k 鈭� y = 0 + k (\frac{1}{2}) = \frac{k}{2}$

Step 3. Calculation

Total volume of solid is determined as follows,

$V = {鈭� 蟺}_{k = 1}^{n} ({(R (y))}^{2} - {(r (y))}^{2}) 鈭� y = {鈭� 蟺}_{k = 1}^{n} ({(5 - g (y_{k}))}^{2} - 1^{2}) \frac{1}{2} = \frac{蟺}{2} {鈭�}_{k = 1}^{n} ({(5 - {y_{k}}^{2})}^{2} - 1^{2}) = \frac{蟺}{2} {鈭�}_{k = 1}^{4} ({(5 - \frac{k^{2}}{4})}^{2} - 1) = \frac{蟺}{2} (({(5 - \frac{1^{2}}{4})}^{2} - 1) + ({(5 - \frac{2^{2}}{4})}^{2} - 1) + ({(5 - \frac{3^{2}}{4})}^{2} - 1) + ({(5 - \frac{4^{2}}{4})}^{2} - 1)) = \frac{蟺}{2} (\frac{345}{16} + \frac{308}{16} + \frac{273}{16} + 15) = \frac{583}{16} 蟺$

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91影视

Consider the region between $f (x) = \sqrt{x}$ and the x-axis on $[0, 4]$ . For each line of rotation given in Exercises 27鈥�30, use four disks or washers based on the given rectangles to approximate the volume of the resulting solid.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

Step 3. Calculation

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91影视

Consider the region between f(x)=xand the x-axis on [0,4]. For each line of rotation given in Exercises 27鈥�30, use four disks or washers based on the given rectangles to approximate the volume of the resulting solid.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Calculation

Step 3. Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Statistics

Logic and Functions

Theoretical and Mathematical Physics

Decision Maths

Calculus

Study anywhere. Anytime. Across all devices.

Consider the region between $f (x) = \sqrt{x}$ and the x-axis on $[0, 4]$ . For each line of rotation given in Exercises 27鈥�30, use four disks or washers based on the given rectangles to approximate the volume of the resulting solid.