The problem is asking for the volume of a solid generated by revolving the given region about the y-axis. The equation of the curve is $y = 3\sqrt{x}$, bounded by $x = 0$ and $y = 6$. To find the volume, the shell method is applied since we are revolving around the y-axis.

Shell Method Formula:

The volume $V$ using the shell method is given by: $V = 2\pi \int_{x_1}^{x_2} x \cdot h(x) \, dx$ Where:

• $h(x)$ is the height of the shell (which corresponds to $y = 3\sqrt{x}$),
• $x$ is the radius of the shell (distance to the axis of revolution).

Step-by-step Solution:

1. Determine the bounds: The curve is $y = 3\sqrt{x}$. We need to find the bounds for $x$. At $y = 6$, we can solve for $x$: $6 = 3\sqrt{x} \quad \Rightarrow \quad x = 4$ So, we integrate from $x = 0$ to $x = 4$.

2. Height of the shell: The height $h(x)$ is just the equation of the curve, $h(x) = 3\sqrt{x}$.

3. Set up the integral: Using the shell method formula, the volume is: $V = 2\pi \int_0^4 x \cdot (3\sqrt{x}) \, dx$ Simplifying the integrand: $V = 6\pi \int_0^4 x^{3/2} \, dx$

4. Compute the integral: $\int x^{3/2} \, dx = \frac{2}{5}x^{5/2}$ So: $V = 6\pi \left[\frac{2}{5}x^{5/2}\right]_0^4 = 6\pi \cdot \frac{2}{5} \cdot (4^{5/2})$ Since $4^{5/2} = (2^2)^{5/2} = 2^5 = 32$, we get: $V = 6\pi \cdot \frac{2}{5} \cdot 32 = 6\pi \cdot \frac{64}{5} = \frac{384\pi}{5}$

Thus, the volume of the solid is $\frac{384\pi}{5}$.

Answer:

The correct choice is (c) $\frac{384\pi}{5}$.

Do you have any questions, or would you like more details on this solution?

Here are 5 related questions for further exploration:

1. How would the setup change if the region were revolved around the x-axis?
2. How do you apply the washer method to solve a similar problem?
3. What is the difference between the disk and shell methods in calculus?
4. How would the volume change if the upper limit of $y$ was increased to $9$?
5. Can this problem be solved using cylindrical coordinates?

Tip: Always carefully assess which axis the rotation is occurring around to determine whether to use the shell or disk method.