Determine the volume of a solid formed by revolving the region bounded by the curve y=4(sqrtx)
, the line y=16
, and the line x=25
 about the line y=15
.

The volume of the solid is ??? 
 units cubed.

The volume of the solid can be determined using the method of cylindrical shells or the method of washers (disks). Since we are revolving around the line \( y = 15 \), and the function is given in terms of \( y \), we'll use the washer method, adapting to the shifted axis of rotation.

### Step-by-Step Solution:

1. **Equation of the curve:**
   \[
   y = 4 \sqrt{x} \implies x = \frac{y^2}{16}
   \]

2. **Bounds of integration:**
   The curve is bounded by \( y = 16 \), so the region to be revolved lies between \( y = 0 \) and \( y = 16 \).

3. **Washer method formula:**
   The general formula for the volume when revolving around \( y = k \) is:
   \[
   V = \pi \int_{a}^{b} \left( R_{\text{outer}}(y)^2 - R_{\text{inner}}(y)^2 \right) dy
   \]
   where \( R_{\text{outer}}(y) \) and \( R_{\text{inner}}(y) \) are the distances from the curve to the axis of revolution.

4. **Outer and inner radii:**
   - The outer radius is from \( y = 16 \) to the line \( y = 15 \). So, \( R_{\text{outer}} = 16 - 15 = 1 \).
   - The inner radius is from the curve \( y = 4\sqrt{x} \), or equivalently \( x = \frac{y^2}{16} \), to the line \( y = 15 \), which is \( R_{\text{inner}} = 15 - y \).

5. **Volume calculation:**
   Now, integrate from \( y = 0 \) to \( y = 16 \):
   \[
   V = \pi \int_{0}^{16} \left(1^2 - (15 - y)^2\right) \, dy
   \]

Let's compute this integral.The volume of the solid is \( -\frac{3328\pi}{3} \) cubic units. The negative sign indicates the setup might need reconsideration regarding the inner and outer radii. Let's refine the calculation by correctly setting the radii terms:

- The inner radius should be from the curve \( y = 15 - y \), and the outer region is fixed at 1.

I'll fix the integration and complete the result.It appears that the inner and outer radii were correctly defined, but the volume formula is yielding a negative result due to a reversed interpretation of the radii. Let's correct the expression and recompute the volume.

We'll calculate using:

\[
V = \pi \int_{0}^{16} \left((15 - y)^2 - 1^2\right) \, dy
\]

Let me recalculate this properly now.The corrected volume of the solid is \( \frac{3328\pi}{3} \) cubic units.

Would you like further details on any step? Here are some related questions:

1. What is the washer method, and when is it used?
2. How do you find the equation of a curve when revolving it around a horizontal line?
3. What is the difference between the shell method and the washer method?
4. Can you explain how to adjust the radii when revolving around non-axis lines?
5. How does shifting the axis of rotation affect the volume calculation?

**Tip:** Always ensure that the outer radius is larger than the inner radius to avoid negative values in volume integrals.

Determine the volume of a solid formed by revolving the region bounded by the curve y=4(sqrtx), the line y=16, and the line x=25 about the line y=15. The volume of the solid is ??? units cubed.

This detailed guide walks through the steps of calculating the volume of a solid formed by revolving the region bounded by the curve y=4√x, the line y=16, and x=25 around y=15. The solution applies the washer method and definite integration, suitable for calculus students at the college level.

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