Consider the blue vertical line shown above (click on graph for better view) connecting the graphs y=g(x)=sin(x) and y=f(x)=cos(2x)

.
Referring to this blue line, match the statements below about rotating this line with the corresponding statements about the result obtained.

1. The result of rotating the line about the 
x-axis is
2. The result of rotating the line about the 
y-axis is
3. The result of rotating the line about the line 
 is y=1
4. The result of rotating the line about the line 
 is x=-2
5. The result of rotating the line about the line 
 is x=pi
6. The result of rotating the line about the line 
 is y=-2
7. The result of rotating the line about the line 
 is y=pi
8. The result of rotating the line about the line 
 is y=-pi

A. a cylinder of radius pi - x
 and height cos(2x) - sin(x)

B. an annulus with inner radius 2 + sin(x)
 and outer radius 2 + cos(2x)

C. an annulus with inner radius pi + sin(x)
 and outer radius pi + cos(2x)

D. an annulus with inner radius pi - cos(2x)
 and outer radius pi - sin(x)

E. an annulus with inner radius 1 - cos(2x)
 and outer radius 1 - sin(x)

F. an annulus with inner radius sin(x)
 and outer radius cos(2x)

G. a cylinder of radius x
 and height cos(2x) - sin(x)

H. a cylinder of radius x + 2
 and height cos(2x) - sin(x)

Let's go through each of the statements and match the corresponding result from the given choices.

1. **The result of rotating the line about the x-axis:**
   - When we rotate a line segment around the x-axis, we create a surface of revolution. Since the line connects \(y = \sin(x)\) and \(y = \cos(2x)\), rotating this line will result in a cylinder with height equal to the vertical distance between the two curves at a given \(x\) and radius corresponding to the x-coordinate.
   - The formula that fits this description is: **G. a cylinder of radius \(x\) and height \(\cos(2x) - \sin(x)\)**.

2. **The result of rotating the line about the y-axis:**
   - Rotating around the y-axis will result in a surface whose cross-section varies with the distance from the origin, essentially stretching the radius by a factor involving \(x\).
   - The correct description is: **H. a cylinder of radius \(x + 2\) and height \(\cos(2x) - \sin(x)\)**.

3. **The result of rotating the line about the line \(y = 1\):**
   - When rotating about \(y = 1\), the radii will be determined by how far the functions \(y = \sin(x)\) and \(y = \cos(2x)\) are from \(y = 1\).
   - The correct answer is: **E. an annulus with inner radius \(1 - \cos(2x)\) and outer radius \(1 - \sin(x)\)**.

4. **The result of rotating the line about the line \(x = -2\):**
   - Rotating about a vertical line like \(x = -2\) will result in an annulus with radii involving \(x + 2\), i.e., shifting the x-coordinates by 2.
   - The correct answer is: **B. an annulus with inner radius \(2 + \sin(x)\) and outer radius \(2 + \cos(2x)\)**.

5. **The result of rotating the line about the line \(x = \pi\):**
   - Rotating about \(x = \pi\) shifts the x-coordinate by \(\pi\), so the radii of the annulus will involve terms like \(\pi - x\).
   - The correct answer is: **A. a cylinder of radius \(\pi - x\) and height \(\cos(2x) - \sin(x)\)**.

6. **The result of rotating the line about the line \(y = -2\):**
   - Rotating around \(y = -2\) shifts the y-values by 2, creating an annulus with radii involving \(-2 + \cos(2x)\) and \(-2 + \sin(x)\).
   - The correct answer is: **F. an annulus with inner radius \(\sin(x)\) and outer radius \(\cos(2x)\)**.

7. **The result of rotating the line about the line \(y = \pi\):**
   - Rotating about \(y = \pi\) shifts the y-values by \(\pi\), which results in an annulus with inner and outer radii shifted by \(\pi\).
   - The correct answer is: **C. an annulus with inner radius \(\pi + \sin(x)\) and outer radius \(\pi + \cos(2x)\)**.

8. **The result of rotating the line about the line \(y = -\pi\):**
   - Similar to the rotation about \(y = \pi\), except the radii are now shifted by \(-\pi\).
   - The correct answer is: **D. an annulus with inner radius \(\pi - \cos(2x)\) and outer radius \(\pi - \sin(x)\)**.

### Final matches:

1. G
2. H
3. E
4. B
5. A
6. F
7. C
8. D

Do you want details or have any questions about these? 

Here are 5 related questions:
1. How would the result change if we rotate about the z-axis?
2. What if the functions involved were \(y = e^x\) and \(y = \ln(x)\)?
3. How is the volume of these surfaces of revolution calculated?
4. How does the distance between the curves affect the final shape?
5. What happens if the line is rotated by an angle before rotating about the axis?

**Tip:** For surfaces of revolution, using the method of cylindrical shells or disks helps calculate volumes easily.

Consider the blue vertical line shown above connecting the graphs y=g(x)=sin(x) and y=f(x)=cos(2x). Referring to this blue line, match the statements below about rotating this line with the corresponding statements about the result obtained. 1. The result of rotating the line about the x-axis is 2. The result of rotating the line about the y-axis is 3. The result of rotating the line about the line y=1 is 4. The result of rotating the line about the line x=-2 is 5. The result of rotating the line about the line x=pi is 6. The result of rotating the line about the line y=-2 is 7. The result of rotating the line about the line y=pi is 8. The result of rotating the line about the line y=-pi is

Explore how rotating a vertical line between y = sin(x) and y = cos(2x) creates various surfaces of revolution, including cylinders and annuli. Understand how different axes of rotation affect the resulting shape. Ideal for advanced high school or undergraduate students.

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