 Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Use Washer method.
y = sqrt(x),   y = 0,   x = 4;  about x = 9 

To set up the integral for the volume of the solid formed by rotating the region bounded by the curves \( y = \sqrt{x} \), \( y = 0 \), and \( x = 4 \) about the line \( x = 9 \) using the Washer method, follow these steps:

### 1. Sketch the region and identify the boundaries
- The region is bounded by \( y = \sqrt{x} \) (a curve), \( y = 0 \) (the x-axis), and \( x = 4 \) (a vertical line).
- The area of interest lies between \( x = 0 \) and \( x = 4 \).

### 2. Formula for the Washer Method
The volume of a solid of revolution using the washer method is given by:
\[
V = \pi \int_{a}^{b} \left[ (R_{\text{outer}}(x))^2 - (R_{\text{inner}}(x))^2 \right] \, dx
\]
where:
- \( R_{\text{outer}}(x) \) is the distance from the axis of rotation to the outer curve,
- \( R_{\text{inner}}(x) \) is the distance from the axis of rotation to the inner curve.

### 3. Determine the radii
- The axis of rotation is \( x = 9 \).
- The outer curve is \( y = \sqrt{x} \). To find the outer radius, we calculate the horizontal distance from the line \( x = 9 \) to the curve \( x = x \). This distance is:
  \[
  R_{\text{outer}}(x) = 9 - x
  \]
- The inner radius corresponds to the distance from \( x = 9 \) to the line \( x = 0 \) (the lower boundary \( y = 0 \)). This distance is:
  \[
  R_{\text{inner}}(x) = 9 - 0 = 9
  \]

### 4. Set up the integral
We integrate with respect to \( x \) from 0 to 4 (the boundaries of \( x \)), and the volume is given by:
\[
V = \pi \int_{0}^{4} \left[ (9 - 0)^2 - (9 - x)^2 \right] \, dx
\]
Simplifying:
\[
V = \pi \int_{0}^{4} \left[ 81 - (9 - x)^2 \right] \, dx
\]

This is the integral that represents the volume of the solid obtained by rotating the region around \( x = 9 \).

Would you like further clarification or details on how this setup works?

### 5 Related Questions:
1. What if the region was rotated about a different vertical line, such as \( x = 5 \)?
2. How would the setup change if the region was rotated around the x-axis instead of \( x = 9 \)?
3. Can the Disk Method be used for this problem, and if so, how?
4. How would we evaluate the integral to compute the actual volume?
5. What happens if the region was rotated around a horizontal line like \( y = 2 \)?

### Tip:
When using the washer method, always carefully calculate the outer and inner radii based on the distance from the axis of rotation to the curve or boundary.

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Use Washer method. y = sqrt(x), y = 0, x = 4; about x = 9.

Learn how to set up the integral for finding the volume of a solid formed by rotating the region bounded by y = sqrt(x), y = 0, and x = 4 about the line x = 9 using the Washer method. This solution covers key calculus concepts, including the setup of the integral and the use of the washer method for volume calculation, suitable for advanced high school and college-level students.

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