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To solve this problem using the Washer Method, we need to find the volume of the solid generated by revolving the specified region around the \( y \)-axis. The region is bounded by:

1. The circle \( x^2 + y^2 = 4 \).
2. The vertical line \( x = 2 \) on the right.
3. The horizontal line \( y = 2 \) above.

The Washer Method formula for finding the volume when revolving about the \( y \)-axis is:
\[
V = \pi \int_{a}^{b} \left[ R(y)^2 - r(y)^2 \right] \, dy
\]
where:
- \( R(y) \) is the outer radius (distance from the \( y \)-axis to the outer edge of the region).
- \( r(y) \) is the inner radius (distance from the \( y \)-axis to the inner edge of the region).
- \( a \) and \( b \) are the bounds for \( y \).

### Step 1: Identify the bounds
The circle \( x^2 + y^2 = 4 \) has a radius of 2, so the \( y \)-values range from \( -2 \) to \( 2 \). Given the upper boundary is \( y = 2 \), the integration bounds will be from \( 0 \) to \( 2 \) (considering the region above the x-axis).

### Step 2: Determine \( R(y) \) and \( r(y) \)
- **Outer radius \( R(y) \)**: The line \( x = 2 \) is the outer boundary, so \( R(y) = 2 \).
- **Inner radius \( r(y) \)**: The equation of the circle can be rewritten as \( x = \sqrt{4 - y^2} \), giving \( r(y) = \sqrt{4 - y^2} \).

### Step 3: Set up the integral
The volume is:
\[
V = \pi \int_{0}^{2} \left[ 2^2 - (\sqrt{4 - y^2})^2 \right] \, dy = \pi \int_{0}^{2} \left[ 4 - (4 - y^2) \right] \, dy
\]
which simplifies to:
\[
V = \pi \int_{0}^{2} y^2 \, dy
\]

### Step 4: Evaluate the integral
\[
V = \pi \left[ \frac{y^3}{3} \right]_{0}^{2} = \pi \left( \frac{2^3}{3} - \frac{0^3}{3} \right) = \pi \left( \frac{8}{3} \right)
\]

Thus, the volume of the solid is:
\[
V = \frac{8\pi}{3}
\]

Would you like more details or have any questions?

### Related Questions:
1. How does the Washer Method differ from the Disk Method?
2. How would the problem change if the region was revolved about the \(x\)-axis?
3. What is the geometric interpretation of \( R(y) \) and \( r(y) \)?
4. How does the integral setup change if the region is bounded by different curves?
5. How would you find the volume using cylindrical shells?

### Tip:
Always sketch the region and the axis of revolution before setting up the integral. It helps visualize \( R(y) \) and \( r(y) \) correctly.

Find the volume of the solid generated by revolving the region bounded on the left by the circle x^2 + y^2 = 4, on the right by the line x = 2, and above by the line y = 2, about the y-axis. (Use the Washer Method).

This calculus problem explores finding the volume of a solid generated by revolving a region around the y-axis using the Washer Method. The region is bounded by the circle x^2 + y^2 = 4, the vertical line x = 2, and the horizontal line y = 2. Step-by-step solutions using integration are provided. Suitable for Grade 12 or college-level calculus students.

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