Answer:
To find the volume of the solid obtained by rotating the region bounded by the curve \(22 + (y - 3)^2 = 4\) about the y-axis, we can use the method of cylindrical shells.
The given curve can be rewritten as:
\[y = 3 \pm \sqrt{4 - (x - 22)}\]
Now, we will rotate the region bounded by the curve about the y-axis. This creates cylindrical shells with radius \(x\) and height \(3 \pm \sqrt{4 - (x - 22)}\).
The volume of each shell is given by:
\[V_{\text{shell}} = 2\pi x \cdot h \cdot \Delta x\]
Where \(h\) is the height of the shell, which is \(3 \pm \sqrt{4 - (x - 22)}\), and \(\Delta x\) is the width of the shell in the x-direction.
To find the limits of integration for \(x\), we set \(4 - (x - 22) \geq 0\), which gives us \(x \leq 18\). So, the limits of integration are \(x = 0\) to \(x = 18\).
Now, we integrate to find the total volume:
\[V = \int_{0}^{18} 2\pi x \cdot \left(3 \pm \sqrt{4 - (x - 22)}\right) \, dx\]
This integral will need to be evaluated twice, once for each term inside the square root. The sum of these integrals will give you the total volume of the solid.
Keep in mind that you'll need to use appropriate integration techniques, such as u-substitution, to evaluate the integrals. Once you've calculated the two integrals and added them together, you'll have the volume of the solid obtained by rotating the given curve about the y-axis.