Set up an integral that represents the volume generated by rotating the region bounded by the curves y = 4x - x^2 and the X-axis about x = 7.

To set up an integral for the volume of the solid generated by rotating the region bounded by \( y = 4x - x^2 \) and the X-axis about the line \( x = 7 \), we can use the **cylindrical shell method**. This method is appropriate when rotating around a vertical axis that is not the y-axis.

### Step-by-step process:
1. **Identify the boundaries**:
   - The curve given is \( y = 4x - x^2 \), and it intersects the X-axis at \( y = 0 \).
   - Set \( y = 4x - x^2 = 0 \) to find the points of intersection:
     \[
     x(4 - x) = 0 \implies x = 0 \quad \text{and} \quad x = 4.
     \]
   - Thus, the region is bounded by \( x = 0 \) and \( x = 4 \) on the x-axis.

2. **Formula for the cylindrical shell method**:
   The volume of a cylindrical shell is given by:
   \[
   V = \int_{a}^{b} 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx
   \]
   where:
   - **Radius** is the distance from \( x \) to the axis of rotation \( x = 7 \), i.e., \( |7 - x| = 7 - x \) (since \( x \leq 7 \)).
   - **Height** is the value of the function, \( y = 4x - x^2 \).

3. **Set up the integral**:
   The integral for the volume is:
   \[
   V = \int_{0}^{4} 2\pi (7 - x)(4x - x^2) \, dx.
   \]

This integral represents the volume of the solid formed by rotating the region about the line \( x = 7 \).

Would you like a step-by-step calculation of this integral, or any clarifications?

Here are some follow-up questions to deepen your understanding:
1. What are the limits of integration for a volume of revolution problem using the disk/washer method instead of the shell method?
2. How does changing the axis of rotation from \( x = 7 \) to \( x = 10 \) affect the volume?
3. What if the curve was \( y = 3x - x^2 \) instead of \( y = 4x - x^2 \)? How would the integral setup change?
4. Can this region be rotated about the y-axis using the disk method instead of shells? How would you set that up?
5. How do you interpret the meaning of "radius" in the context of cylindrical shells?

**Tip**: In the shell method, always think of the radius as the horizontal distance from the curve to the axis of rotation!

Learn how to set up an integral using the cylindrical shell method to calculate the volume generated by rotating the region bounded by y = 4x - x^2 and the X-axis about the line x = 7. This problem covers integral calculus concepts and is suitable for advanced high school or undergraduate 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