Find the volume *V* of the solid obtained by rotating the region bounded by the given curves about the specified line.

*x* = 7*y*2,  *x* = 7;  about *x* = 7


To find the volume of the solid obtained by rotating the region bounded by the curves \( x = 7y^2 \) and \( x = 7 \) about the line \( x = 7 \), we will use the **method of washers** since we are rotating around a vertical line.

### Step-by-Step Solution

1. **Sketch the Region:**

   The curve \( x = 7y^2 \) is a parabola that opens to the right, and \( x = 7 \) is a vertical line. The region bounded by these two curves lies between the parabola and the vertical line.

2. **Determine the Points of Intersection:**

   To find the points where the curves intersect, set:

   \[
   7y^2 = 7
   \]

   Solving for \( y \):

   \[
   y^2 = 1 \implies y = \pm 1
   \]

   Thus, the curves intersect at \( y = -1 \) and \( y = 1 \).

3. **Set Up the Volume Integral Using Washers:**

   When rotating around the line \( x = 7 \), the volume of the solid can be found using the washer method. The formula for the volume of a solid of revolution using washers is:

   \[
   V = \int_{a}^{b} \pi \left[ (\text{Outer Radius})^2 - (\text{Inner Radius})^2 \right] \, dy
   \]

   For this problem:

   - The **Outer Radius** is the distance from the line \( x = 7 \) to the line \( x = 7 \), which is zero:  
     \[
     \text{Outer Radius} = 0
     \]

   - The **Inner Radius** is the distance from the line \( x = 7 \) to the curve \( x = 7y^2 \):

     \[
     \text{Inner Radius} = 7 - 7y^2
     \]

4. **Compute the Volume Integral:**

   Substituting these values into the integral, we get:

   \[
   V = \int_{-1}^{1} \pi \left[ (7 - 7y^2)^2 - 0^2 \right] \, dy
   = \pi \int_{-1}^{1} (7 - 7y^2)^2 \, dy
   \]

   Simplify the expression inside the integral:

   \[
   (7 - 7y^2)^2 = (7 \times 1 - 7y^2) = 49 - 7^2 \times y^4 
   = 
   
### Final Answer:

Would you like more details on any step, or have any questions?

### Additional Questions to Explore:
1. How does the method of cylindrical shells differ from the method of disks/washers for finding volumes of revolution?
2. How would the volume change if the region was rotated about the y-axis instead?
3. What are some real-world applications of finding volumes of solids of revolution?
4. How would the integral setup change if the rotation was about a line other than \( x = 4 \)?
5. What if the curves were different, say \( y = x^2 \) and \( y = \sqrt{x} \)? How would that affect the volume calculation?



Learn how to calculate the volume of a solid obtained by rotating the region bounded by x = 7y^2 and x = 7 about the line x = 7. Detailed step-by-step solution using the method of washers in integral calculus.

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