Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded on the right by the graph of g(y)=8/y and on the left by the y-axis for 4≤y≤6, about the x-axis. Round your answer to the nearest hundredth position.

We can use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the graph of \( g(y) = \frac{8}{y} \) (on the right), the \( y \)-axis (on the left), and between \( y = 4 \) and \( y = 6 \), about the \( x \)-axis.

### Step-by-Step Approach:

1. **Formula for volume using cylindrical shells**:
   The formula for the volume using cylindrical shells is:

   \[
   V = 2\pi \int_{a}^{b} \text{radius} \times \text{height} \, dy
   \]
   
   - The **radius** is the distance from the axis of rotation (the \( x \)-axis) to a point on the curve, which is \( y \).
   - The **height** of each cylindrical shell is given by the function \( g(y) = \frac{8}{y} \).
   
   So, the formula becomes:

   \[
   V = 2\pi \int_{4}^{6} y \cdot \frac{8}{y} \, dy
   \]

2. **Simplifying the integral**:
   Notice that \( y \cdot \frac{8}{y} = 8 \), so the integral simplifies to:

   \[
   V = 2\pi \int_{4}^{6} 8 \, dy
   \]

3. **Evaluate the integral**:
   Now, we integrate:

   \[
   \int_{4}^{6} 8 \, dy = 8(y) \Big|_4^6 = 8(6 - 4) = 8 \times 2 = 16
   \]

4. **Final volume**:
   Multiply by \( 2\pi \):

   \[
   V = 2\pi \times 16 = 32\pi
   \]

5. **Approximating the result**:
   Now, approximate \( V \):

   \[
   V \approx 32 \times 3.1416 = 100.53
   \]

### Final Answer:
The volume of the solid is approximately \( \boxed{100.53} \) cubic units.

Would you like further details or clarification? Here are some related questions:

1. How does the method of cylindrical shells differ from the disk method?
2. What happens if the region is rotated about the y-axis instead?
3. How do we handle problems where the bounds are more complex?
4. Can we use cylindrical shells if the function was instead \( g(y) = \frac{y}{8} \)?
5. How would the solution change if the limits were from 3 to 5?

**Tip**: When solving problems using cylindrical shells, always check if the integral simplifies before performing complex operations—it can save time!

Use cylindrical shells to find the volume of the solid obtained by rotating the region bounded on the right by the graph of g(y)=8/y and on the left by the y-axis for 4≤y≤6, about the x-axis. Round your answer to the nearest hundredth position.

Learn how to calculate the volume of a solid formed by rotating the region bounded by g(y) = 8/y and the y-axis between y = 4 and y = 6, using the cylindrical shells method. This problem covers key integral calculus concepts, including volume of solids of revolution. Suitable for advanced high school or college-level students.

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