https://file.aiquickdraw.com/mathbot/file/1726941957641rv4s5l86.jpg

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

### Step-by-Step Solution

1. **Set up the problem:**
   The region is bounded by the curves \( x = 3y^2 \) (a parabola) and \( x = 3 \) (a vertical line). We are rotating this region around the line \( x = 3 \).

2. **Cylindrical shells:**
   For the cylindrical shells method, the radius of each shell is the distance from a point on the curve \( x = 3y^2 \) to the axis of rotation \( x = 3 \). 
   
   The height of each shell is given by the difference in the \( x \)-coordinates: 
   \[
   h(y) = 3 - 3y^2
   \]

3. **Volume integral:**
   The formula for the volume using cylindrical shells is:
   \[
   V = \int_{a}^{b} 2\pi (\text{radius}) \times (\text{height}) \, dy
   \]
   Here, the radius is \( r(y) = 3 - (3y^2) \) and the height is the \( y \)-length of the region.
   
   The bounds are from \( y = 0 \) to \( y = 1 \) because \( x = 3y^2 \) intersects \( x = 3 \) when \( y = 1 \).
   
4. **Setup the integral:**
   \[
   V = 2\pi \int_0^1 (3 - 3y^2)(1) \, dy
   \]
   
5. **Evaluate the integral:**
   \[
   V = 2\pi \int_0^1 (3 - 3y^2) \, dy
   \]
   First, integrate each term:
   \[
   \int_0^1 3 \, dy = 3y \Big|_0^1 = 3
   \]
   \[
   \int_0^1 3y^2 \, dy = y^3 \Big|_0^1 = 1
   \]
   
6. **Final calculation:**
   \[
   V = 2\pi \left( 3 - 1 \right) = 2\pi \times 2 = 4\pi
   \]

Thus, the volume of the solid is \( V = 4\pi \).

Would you like further details or clarifications on any of the steps?

Here are five related questions for further exploration:
1. How would the result change if the rotation were about the \( y \)-axis?
2. What if the region were rotated around the line \( x = 0 \) instead?
3. How do you calculate the volume using the disk method for this same problem?
4. What happens if the parabola is \( x = ky^2 \) for some constant \( k \)?
5. How can we generalize this technique for other curves?

**Tip:** The cylindrical shells method is particularly useful when the region is rotated around a vertical line and the slices would be awkward to express in terms of \( x \).

Consider the solid obtained by rotating the region bounded by the given curves about the line x = 3. The curves are x = 3y^2 and x = 3. Find the volume V of this solid.

Learn how to find the volume of the solid obtained by rotating the region bounded by the curves x = 3y^2 and x = 3 around the line x = 3 using the cylindrical shells method. This problem involves setting up a definite integral and calculating the volume of revolution. Suitable for Grades 11-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation