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This problem involves finding the volume of a solid generated by rotating a region about the \(y\)-axis using the method of cylindrical shells.

### Part (a): Setting up the integral

The equation provided is \( y = 2\sqrt{x} - 3 \). To express \( x \) as a function of \( y \), solve for \( x \) in terms of \( y \):

\[
y = 2\sqrt{x} - 3
\]

First, add 3 to both sides:

\[
y + 3 = 2\sqrt{x}
\]

Now, divide by 2:

\[
\frac{y + 3}{2} = \sqrt{x}
\]

Finally, square both sides to isolate \( x \):

\[
x = \left(\frac{y + 3}{2}\right)^2
\]

So the function is \( x = \frac{(y + 3)^2}{4} \).

Next, set up the integral for the volume using the formula for the volume of a solid of revolution around the \(y\)-axis:

\[
V = 2\pi \int_{a}^{b} x(y) \cdot dy
\]

For this problem, the limits of integration are from \( y = 0 \) to \( y = 3 \). The function for \( x \) is \( \frac{(y + 3)^2}{4} \).

Thus, the definite integral becomes:

\[
V = 2\pi \int_{0}^{3} \frac{(y + 3)^2}{4} \, dy
\]

### Part (b): Calculating the volume

Now, I'll calculate the volume based on this integral.

Let me evaluate the integral and calculate the volume.The volume of the solid generated by rotating the region around the \( y \)-axis is approximately \( 98.960169 \, \text{units}^3 \), rounded to six decimal places.

Would you like a detailed breakdown or further clarification?

Here are five related questions for further practice:
1. How would you set up the integral if the region was rotated about the \(x\)-axis instead?
2. What would change in the problem if the equation was \( y = 2\sqrt{x} - 4 \) instead of \( y = 2\sqrt{x} - 3 \)?
3. How would you handle the problem if the limits of integration were different, say from \(y = 1\) to \(y = 3\)?
4. Can you derive the cylindrical shell formula for rotation about the \(x\)-axis?
5. What happens to the volume if the function is scaled by a constant?

**Tip**: When solving volume problems involving rotations, always verify whether you're rotating around the \(x\)-axis or \(y\)-axis to choose the right method (disks, washers, or cylindrical shells).

Find the volume of the solid obtained by rotating the region bounded by the given curves about the y-axis. y = 2√x - 3, y = 0, y = 3. Part a) Set up a definite integral that gives the volume. Part b) Find the volume of the solid. If necessary, round to six decimal places.

Find the volume of a solid generated by rotating the region bounded by y = 2√x - 3, y = 0, and y = 3 around the y-axis. Set up a definite integral and solve using the cylindrical shell method. Step-by-step solution and final volume calculation provided.

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