Let R
 be the region bounded by y=x^2
, x=0
, and y=4
. Set up the integral that represents the volume of the solid formed when R
 is revolved about the line x=−2
 using the Shell Method.

To solve this problem using the **Shell Method**, we need to revolve the region \( R \) around the line \( x = -2 \) and set up the corresponding integral.

### Step-by-Step Setup:

1. **Region Definition:**
   The region \( R \) is bounded by the parabola \( y = x^2 \), the vertical line \( x = 0 \), and the horizontal line \( y = 4 \). This implies that the limits for \( x \) are from 0 to \( 2 \) because \( x = 2 \) when \( y = 4 \).

2. **Shell Method Formula:**
   The volume of a solid of revolution using the Shell Method is given by:
   \[
   V = 2\pi \int_{a}^{b} \text{(radius)} \times \text{(height)} \, dx
   \]
   
3. **Radius and Height:**
   - The **radius** of a shell at any point \( x \) is the horizontal distance from the line \( x = -2 \). The distance between \( x \) and \( x = -2 \) is \( x - (-2) = x + 2 \).
   - The **height** of a shell at any point \( x \) is the value of \( y = x^2 \), which represents the height of the region up to the curve.

4. **Integral Setup:**
   The volume of the solid is therefore given by:
   \[
   V = 2\pi \int_{0}^{2} (x + 2) \cdot (4 - x^2) \, dx
   \]
   where \( (x + 2) \) is the radius and \( (4 - x^2) \) is the height.

This integral represents the volume of the solid formed when the region \( R \) is revolved about the line \( x = -2 \).

### Would you like me to solve this integral or provide more details about the steps?

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Here are 5 related questions to expand your understanding:
1. How would the integral change if the region was revolved around the line \( x = 1 \)?
2. What is the geometric interpretation of the Shell Method in terms of slices of the solid?
3. Can you set up a similar integral using the Disk Method for this same problem?
4. How does the height of the shells change as \( x \) varies from 0 to 2?
5. What would be the effect on the volume if the region were revolved around the y-axis instead of \( x = -2 \)?

**Tip:** The Shell Method is particularly useful when revolving a region around a vertical axis that is not the y-axis. The radius is usually a horizontal distance in such cases.

Let R be the region bounded by y = x^2, x = 0, and y = 4. Set up the integral that represents the volume of the solid formed when R is revolved about the line x = -2 using the Shell Method.

Learn how to set up the integral using the Shell Method for finding the volume of a solid formed when the region bounded by y = x^2, x = 0, and y = 4 is revolved around the line x = -2. This problem explores key concepts in integral calculus, including shells, parabolas, and solids of revolution.

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