The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.
x^2 + (y − 2)^2 = 4;
   about the y-axis

The given curve is the equation of a circle:

\[
x^2 + (y - 2)^2 = 4
\]

This represents a circle with a radius of 2, centered at \((0, 2)\). We are asked to find the volume of the solid formed by rotating this region about the **y-axis**.

### Step 1: Rewriting the equation of the circle

To simplify, solve for \(x^2\):

\[
x^2 = 4 - (y - 2)^2
\]

This will be used to set up the integral.

### Step 2: Determine the bounds for \(y\)

The circle's radius is 2, and its center is at \(y = 2\). So the \(y\)-coordinates range from:

\[
y = 0 \quad \text{to} \quad y = 4
\]

### Step 3: Set up the integral using the **disk method**

Since we are rotating around the y-axis, we use the disk method. The radius of each disk at a particular \(y\) is given by \(x = \sqrt{4 - (y - 2)^2}\). The area of a disk is \(\pi r^2\), where \(r = x\).

The volume is given by the integral:

\[
V = \pi \int_{0}^{4} \left[ 4 - (y - 2)^2 \right] dy
\]

### Step 4: Solve the integral

First, expand \((y - 2)^2\):

\[
(y - 2)^2 = y^2 - 4y + 4
\]

So the integrand becomes:

\[
4 - (y^2 - 4y + 4) = 4 - y^2 + 4y - 4 = -y^2 + 4y
\]

Now, integrate:

\[
V = \pi \int_0^4 (-y^2 + 4y) \, dy
\]

\[
V = \pi \left[ \frac{-y^3}{3} + 2y^2 \right]_0^4
\]

Evaluate the integral at the bounds \(y = 0\) and \(y = 4\):

At \(y = 4\):

\[
\frac{-4^3}{3} + 2(4^2) = \frac{-64}{3} + 2(16) = \frac{-64}{3} + 32 = \frac{-64 + 96}{3} = \frac{32}{3}
\]

At \(y = 0\):

\[
\frac{-0^3}{3} + 2(0^2) = 0
\]

Thus, the volume is:

\[
V = \pi \left( \frac{32}{3} - 0 \right) = \frac{32\pi}{3}
\]

### Final Answer:
The volume of the solid is:

\[
V = \frac{32\pi}{3} \, \text{cubic units}
\]

---

Would you like further details or explanations on any part of this process?

Here are some related questions to expand your understanding:
1. How would the method change if the curve were rotated about the x-axis?
2. What if the region was rotated around a line other than the x or y-axis?
3. Can you derive the formula for the volume of a sphere using similar methods?
4. How does the shell method compare to the disk method for this problem?
5. What happens if the radius of the circle was different?

**Tip:** When using the disk or washer method, always carefully identify the radius of the disks or washers from the given equation to avoid errors.

Learn how to calculate the volume of a solid formed by rotating the region bounded by x^2 + (y - 2)^2 = 4 about the y-axis. This problem involves the disk method, integral calculus, and the geometry of circles. Suitable for students in Grades 11-12.

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