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The problem asks to find the region cut from the solid elliptical cylinder \(x^2 + 4y^2 \leq 4\) by the xy-plane and the plane \(z = x + 2\).

### Let's break this down:

1. **Elliptical Cylinder:**
   - The inequality \(x^2 + 4y^2 \leq 4\) represents an elliptical cylinder aligned along the z-axis. 
   - The cross-section in the xy-plane is an ellipse with semi-major axis of 2 along the x-axis and semi-minor axis of 1 along the y-axis.

2. **Plane \(z = x + 2\):**
   - This plane intersects the elliptical cylinder at different points, depending on the values of \(x\) and \(y\).

### To find the volume of the region cut out:
We will integrate over the region defined by the ellipse in the xy-plane and use the plane equation \(z = x + 2\) as the height.

The boundaries for \(x\) and \(y\) are determined by the elliptical equation \(x^2 + 4y^2 = 4\).

### Steps:

1. **Express the region in terms of \(x\) and \(y\):**
   - Solve for \(y\) bounds in terms of \(x\): From the equation \(x^2 + 4y^2 = 4\),
     \[
     4y^2 = 4 - x^2 \quad \Rightarrow \quad y^2 = 1 - \frac{x^2}{4} \quad \Rightarrow \quad y = \pm \sqrt{1 - \frac{x^2}{4}}
     \]
   - This defines the bounds for \(y\) for each \(x\).

2. **Set up the volume integral:**
   The volume is given by the integral:
   \[
   V = \int_{-2}^{2} \int_{-\sqrt{1 - \frac{x^2}{4}}}^{\sqrt{1 - \frac{x^2}{4}}} (x + 2) \, dy \, dx
   \]
   where the height \(z = x + 2\) is used.

   Now we can compute the volume using this double integral.

Would you like me to continue solving this integral in detail or clarify any part so far? 

Here are 5 questions related to the problem:
1. How do we interpret the bounds for \(x\) and \(y\) in the elliptical equation?
2. Why is the plane equation \(z = x + 2\) used as the height in the volume integral?
3. How does the cross-section of the elliptical cylinder influence the limits of integration?
4. What role does the shape of the region play in determining the limits of integration?
5. How can changing the plane equation (e.g., \(z = x + 3\)) affect the solution?

**Tip:** When working with volumes of solids bounded by planes, always visualize the region and sketch the boundaries. This helps in setting up the correct limits of integration.

The region cut from the solid elliptical cylinder x^2 + 4y^2 <= 4 by the xy-plane and the plane z = x + 2

Learn how to find the volume of a region cut from a solid elliptical cylinder by the xy-plane and a given plane using double integration. This problem covers key multivariable calculus concepts such as setting up integrals and using the equation of an ellipse for determining bounds. Suitable for university-level calculus students.

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