Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
x = 2*(sqrt(3y))
,   x = 0,   y = 7;
   about the y-axis

To find the volume of the solid obtained by rotating the region bounded by the curves \( x = 2\sqrt{3y} \), \( x = 0 \), and \( y = 7 \) about the y-axis, we use the method of **cylindrical shells**. This method is particularly useful when rotating around a vertical axis like the y-axis.

### Step-by-Step Solution

1. **Identify the bounds of integration**:
   - The region is bounded by \( y \) from \( y = 0 \) to \( y = 7 \).
   - The curve is \( x = 2\sqrt{3y} \).

2. **Set up the volume integral using the shell method**:
   The formula for the volume using cylindrical shells when rotating around the y-axis is:

   \[
   V = 2\pi \int_{a}^{b} \text{(radius)} \times \text{(height)} \, dy
   \]

   - **Radius**: The distance from the y-axis to the shell, which is just \( x = 2\sqrt{3y} \).
   - **Height**: The height of the shell is the function \( x = 2\sqrt{3y} \).
   - The bounds for \( y \) are from \( 0 \) to \( 7 \).

   Therefore, the integral is:

   \[
   V = 2\pi \int_{0}^{7} (2\sqrt{3y}) \times (2\sqrt{3y}) \, dy
   \]

   Simplifying the integrand:

   \[
   V = 2\pi \int_{0}^{7} 4 \times 3y \, dy = 24\pi \int_{0}^{7} y \, dy
   \]

3. **Evaluate the integral**:

   \[
   V = 24\pi \left[ \frac{y^2}{2} \right]_{0}^{7} = 24\pi \times \left( \frac{7^2}{2} - \frac{0^2}{2} \right)
   \]

   \[
   V = 24\pi \times \frac{49}{2} = 24\pi \times 24.5 = 588\pi
   \]

### Final Answer:

The volume of the solid is \( V = 588\pi \) cubic units.

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Do you want more details or have any questions?

Here are 5 related questions:

1. How do you determine whether to use the disk/washer method or the shell method for volume?
2. What would change if the region was rotated about the x-axis instead of the y-axis?
3. How would the volume calculation differ if the bounds for \( y \) were different?
4. Can you find the volume using the disk method instead of the shell method?
5. How would the formula change if the curve was \( x = \sqrt{y} \) instead of \( x = 2\sqrt{3y} \)?

**Tip:** When solving volume problems involving rotation, it's crucial to sketch the region and identify the axis of rotation to determine the most suitable method.

Learn how to find the volume of the solid obtained by rotating the region bounded by x = 2(sqrt(3y)), x = 0, and y = 7 about the y-axis using cylindrical shells method. Detailed step-by-step solution included. Suitable for advanced high school students and beyond.

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