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:

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

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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.