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To solve this problem, we need to set up two integrals to find the volume of the region between the curves \( y = 4x \) and \( y = 2x^2 - 2x + 4 \) when rotated about the \( x \)-axis. We will use the **Washer Method** and the **Shell Method**.

### Step 1: Find the Intersection Points
To determine the limits of integration, we must find the intersection points of the curves by setting:
\[
4x = 2x^2 - 2x + 4.
\]

Rearrange this equation to:
\[
2x^2 - 6x + 4 = 0,
\]
which simplifies to
\[
x^2 - 3x + 2 = 0.
\]

This factors as:
\[
(x - 1)(x - 2) = 0.
\]
So, the solutions are \( x = 1 \) and \( x = 2 \).

These are the limits for both integrals.

### (1) Washer Method
For the **Washer Method**, we consider the volume of washers formed by rotating the region around the \( x \)-axis. The formula for volume using washers is:
\[
V = \pi \int_{a}^{b} \left( R(x)^2 - r(x)^2 \right) \, dx,
\]
where \( R(x) \) is the outer radius and \( r(x) \) is the inner radius.

In this problem:
- The outer radius \( R(x) \) is the distance from the \( x \)-axis to the curve \( y = 2x^2 - 2x + 4 \).
- The inner radius \( r(x) \) is the distance from the \( x \)-axis to the curve \( y = 4x \).

Thus, we have:
\[
R(x) = 2x^2 - 2x + 4 \quad \text{and} \quad r(x) = 4x.
\]

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

### (2) Shell Method
For the **Shell Method**, we consider cylindrical shells created by rotating the region around the \( x \)-axis. The formula for volume using shells is:
\[
V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx.
\]

In this case:
- The radius of each shell is \( y \), which ranges from \( y = 4x \) to \( y = 2x^2 - 2x + 4 \).
- The height of each shell, for a fixed \( y \), is the horizontal distance between the curves in terms of \( x \). We solve each equation for \( x \) in terms of \( y \) to express this height.

For \( y = 4x \):
\[
x = \frac{y}{4}.
\]

For \( y = 2x^2 - 2x + 4 \):
Solve this for \( x \) in terms of \( y \), which might be complex. Therefore, setting up the Shell Method integral directly may not be straightforward.

However, if we proceed with \( x \)-integration, then we note

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.

Learn how to find the volume of the solid obtained by rotating the region between the curves y = 4x and y = 2x^2 - 2x + 4 around the x-axis. This problem covers using the Washer and Shell Methods to set up the appropriate integrals for calculating the volume.

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