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solve for washer and shell method for each

To solve this problem, we need to set up two integrals for the volume of the solid formed by rotating the region between the curves \( y = 4x \) and \( y = 2x^2 - 2x + 4 \) about the \( x \)-axis. We’ll approach this using both the washer and shell methods.

### Step 1: Find the Intersection Points
First, let's find the points where the curves intersect by setting \( y = 4x \) equal to \( y = 2x^2 - 2x + 4 \):
\[
4x = 2x^2 - 2x + 4
\]
Rearrange this to:
\[
2x^2 - 6x + 4 = 0
\]
Divide by 2:
\[
x^2 - 3x + 2 = 0
\]
Factor:
\[
(x - 1)(x - 2) = 0
\]
Thus, \( x = 1 \) and \( x = 2 \). These are the \( x \)-values where the curves intersect. To find the corresponding \( y \)-values, substitute \( x = 1 \) and \( x = 2 \) into either equation (e.g., \( y = 4x \)):

- For \( x = 1 \): \( y = 4(1) = 4 \).
- For \( x = 2 \): \( y = 4(2) = 8 \).

So the region of integration for both methods is from \( x = 1 \) to \( x = 2 \).

### Washer Method
When using the washer method, we integrate along the \( x \)-axis. The outer and inner radii are based on the distance from the \( x \)-axis to each curve.

1. **Outer Radius**: \( R(x) = 2x^2 - 2x + 4 \)
2. **Inner Radius**: \( r(x) = 4x \)

The volume of the solid is given by:
\[
V = \pi \int_{1}^{2} \left( R(x)^2 - r(x)^2 \right) \, dx
\]
Substituting the expressions for \( R(x) \) and \( r(x) \):
\[
V = \pi \int_{1}^{2} \left( (2x^2 - 2x + 4)^2 - (4x)^2 \right) \, dx
\]

### Shell Method
For the shell method, we integrate along the \( y \)-axis. This requires expressing \( x \) in terms of \( y \).

1. For \( y = 4x \): \( x = \frac{y}{4} \).
2. For \( y = 2x^2 - 2x + 4 \), solving for \( x \) in terms of \( y \) is more complex. Instead, we use shells based on height differences at each \( x \)-value from 1 to 2.

The height of each shell is the difference in \( y \)-values:
\[
\text{Height} = (2x^2 - 2x + 4) - 4x
\]
The radius of each shell is \( x \).

Thus, the volume integral is:
\[
V = 2\pi \int_{1}^{2} x \left( (2x^2 - 2x + 4) - 4x \right) \, dx
\]

### Final Integrals
To summarize:

1. **Washer Method**:
   \[
   V = \pi \int_{1}^{2} \left( (2x^2 - 2x + 4)^2 - (4x)^2 \right) \, dx
   \]

2. **Shell Method**:
   \[
   V = 2\pi \int_{1}^{2} x \left( (2x^2 - 2x + 4) - 4x \right) \, dx
   \]

These integrals represent the volume of the solid, and as per the instructions, they are not simplified or evaluated.

Would you like more details on setting up either of these methods? 

### Related Questions
1. How do you find the intersection points of two curves?
2. Why is the washer method suitable for rotations around the \( x \)-axis?
3. How would these integrals change if the rotation were around the \( y \)-axis?
4. What are some key differences between the washer and shell methods?
5. Can you derive the formula for volume using cylindrical shells?

### Tip
When setting up volumes of rotation, visualize which radius (outer vs. inner) applies to each curve at a given \( x \)-interval to avoid integration errors.

Consider the region between the curves y = 4x and y = 2x^2 - 2x + 4. Write down two integrals, one involving washers and the other involving shells, to find the volume when this region is rotated about the x-axis to produce a solid. Set up the integrals, but do not simplify or evaluate them.

Explore how to set up integrals using both the washer and shell methods for the region between the curves y = 4x and y = 2x^2 - 2x + 4. This problem demonstrates key calculus concepts in finding volumes of solids of revolution, suitable for undergraduate students.

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