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To solve this problem using the method of cylindrical shells, we will break down the process step-by-step.

### **Given:**
- Curves: \( y = \sqrt{x} \) and \( x = 4y \)
- Axis of rotation: \( x = 17 \)

### **Step 1: Find the Bounds of Integration**
First, let's find the points where the curves intersect:
1. Express \( y = \sqrt{x} \) in terms of \( x \): \( x = y^2 \).
2. Set \( x = y^2 \) equal to \( x = 4y \):
   \[
   y^2 = 4y
   \]
3. Solve for \( y \):
   \[
   y^2 - 4y = 0
   \]
   \[
   y(y - 4) = 0
   \]
   So, \( y = 0 \) and \( y = 4 \).

These values correspond to \( x \)-coordinates \( x = 0 \) (when \( y = 0 \)) and \( x = 16 \) (when \( y = 4 \)).

### **Step 2: Set Up the Cylindrical Shells**
The formula for the volume of a solid of revolution using cylindrical shells is:
\[
V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dy
\]

### **Step 3: Define the Radius and Height of the Shells**
- **Radius:** The radius of a shell at a given \( y \) is the distance from the axis of rotation (\( x = 17 \)) to the \( x \)-coordinate of the shell. For a given \( y \), the \( x \)-coordinate on the curve \( y = \sqrt{x} \) is \( x = y^2 \). Thus, the radius is \( 17 - y^2 \).
- **Height:** The height of the shell at a given \( y \) is the difference in the \( x \)-values of the bounding curves. For a fixed \( y \), the right boundary is \( x = 4y \) and the left boundary is \( x = y^2 \). Therefore, the height is \( 4y - y^2 \).

### **Step 4: Set Up the Integral**
The integral for the volume is:
\[
V = 2\pi \int_{0}^{4} (17 - y^2)(4y - y^2) \, dy
\]

### **Step 5: Simplify and Integrate**
1. Expand the integrand:
   \[
   (17 - y^2)(4y - y^2) = 68y - 17y^3 - 4y^3 + y^5
   \]
   Simplifying this, we get:
   \[
   68y - 21y^3 + y^5
   \]

2. Integrate term by term:
   \[
   \int_{0}^{4} 68y - 21y^3 + y^5 \, dy = \left[ 34y^2 - \frac{21y^4}{4} + \frac{y^6}{6} \right]_{0}^{4}
   \]

3. Substitute \( y = 4 \) into the expression:
   \[
   34(4)^2 - \frac{21(4)^4}{4} + \frac{(4)^6}{6} = 34 \times 16 - \frac{21 \times 256}{4} + \frac{4096}{6}
   \]
   \[
   = 544 - 1344 + \frac{4096}{6}
   \]
   Calculate each term:
   \[
   544 - 1344 + 682.67 \approx -800 + 682.67 = -117.33
   \]

Therefore, the total volume

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis: y = sqrt(x), x = 4y; about x = 17.

Learn how to find the volume of a solid generated by rotating a region bounded by y = sqrt(x) and x = 4y about the line x = 17 using the method of cylindrical shells. This detailed solution covers key calculus concepts, including setting up and solving integrals for volume.

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