Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = 27x^3,  y = 0,  x = 1;  about x = 2

To find the volume of the solid obtained by rotating the region bounded by the curves \( y = 27x^3 \), \( y = 0 \), and \( x = 1 \) about the line \( x = 2 \), we can use the method of cylindrical shells. 

### Step-by-Step Solution:

1. **Set up the problem:**
   - The region is bounded by \( y = 27x^3 \), \( y = 0 \), and \( x = 1 \).
   - We are rotating about the vertical line \( x = 2 \).

2. **Formula for the volume using cylindrical shells:**
   The volume \( V \) is given by the integral:
   \[
   V = 2\pi \int_{a}^{b} ( \text{radius of the shell} ) \times ( \text{height of the shell} ) \, dx
   \]
   Here:
   - The **radius of the shell** is the horizontal distance from the line of rotation \( x = 2 \) to a point on the curve \( y = 27x^3 \), which is \( (2 - x) \).
   - The **height of the shell** is the function \( y = 27x^3 \).

3. **Set up the integral:**
   The limits of integration are from \( x = 0 \) to \( x = 1 \) (since the region is bounded by \( x = 1 \)).

   \[
   V = 2\pi \int_{0}^{1} (2 - x) \times 27x^3 \, dx
   \]
   Simplify the integrand:
   \[
   V = 2\pi \int_{0}^{1} 27x^3(2 - x) \, dx
   \]
   \[
   V = 2\pi \int_{0}^{1} (54x^3 - 27x^4) \, dx
   \]

4. **Evaluate the integral:**
   Integrate term by term:
   \[
   V = 2\pi \left[ \int_{0}^{1} 54x^3 \, dx - \int_{0}^{1} 27x^4 \, dx \right]
   \]
   \[
   V = 2\pi \left[ 54 \cdot \frac{x^4}{4} \bigg|_0^1 - 27 \cdot \frac{x^5}{5} \bigg|_0^1 \right]
   \]
   Evaluate at the limits:
   \[
   V = 2\pi \left[ 54 \cdot \frac{1^4}{4} - 27 \cdot \frac{1^5}{5} \right]
   \]
   \[
   V = 2\pi \left[ 54 \cdot \frac{1}{4} - 27 \cdot \frac{1}{5} \right]
   \]
   Simplify:
   \[
   V = 2\pi \left[ \frac{54}{4} - \frac{27}{5} \right]
   \]
   \[
   V = 2\pi \left[ 13.5 - 5.4 \right]
   \]
   \[
   V = 2\pi \times 8.1
   \]
   \[
   V = 16.2\pi
   \]

### Final Answer:
The volume of the solid is \( \boxed{16.2\pi} \) cubic units.

Would you like more details on this solution, or do you have any further questions?

Here are some related questions:
1. How do you find the volume of a solid using the disk method?
2. What is the difference between the disk and shell methods?
3. How do you find the volume of a solid formed by rotating about a horizontal line?
4. How does the axis of rotation affect the integral setup for volume?
5. What are some common mistakes to avoid when setting up volume integrals?

**Tip:** When dealing with rotation about a vertical line that is not the y-axis, make sure to adjust the radius of the cylindrical shells accordingly.

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 27x^3, y = 0, x = 1; about x = 2

Explore the detailed step-by-step solution for finding the volume of a solid obtained by rotating the curve y = 27x^3 about the line x = 2. Learn how to apply the cylindrical shells method, a core concept in calculus, to solve this problem. Ideal for undergraduate students in calculus courses.

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