Set up the integral that uses the method of disks/washers to find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = 
1/16 *x^2,  x = 5,  y = 0;    about the y-axis

To find the volume \( V \) of the solid obtained by rotating the region bounded by the curves \( y = \frac{1}{16}x^2 \), \( x = 5 \), and \( y = 0 \) about the y-axis, we can use the method of disks or washers.

### Step 1: Express \( x \) as a function of \( y \)

Given \( y = \frac{1}{16}x^2 \), solve for \( x \) in terms of \( y \):
\[
x^2 = 16y
\]
\[
x = \sqrt{16y} = 4\sqrt{y}
\]

### Step 2: Set up the integral using the disk/washer method

Since we are rotating around the y-axis, we will consider vertical slices of the region. Each slice at a given \( y \) will have a radius \( R(y) = x = 4\sqrt{y} \) and a thickness \( dy \). 

The volume element \( dV \) is the volume of a disk with this radius:
\[
dV = \pi [R(y)]^2 \, dy = \pi (4\sqrt{y})^2 \, dy = \pi \cdot 16y \, dy
\]

### Step 3: Determine the limits of integration

The limits of integration correspond to the y-values where the region is bounded. Since the curve \( y = \frac{1}{16}x^2 \) ranges from \( y = 0 \) to \( y \) at \( x = 5 \):
\[
y = \frac{1}{16}(5^2) = \frac{25}{16}
\]
So, the limits of integration are from \( y = 0 \) to \( y = \frac{25}{16} \).

### Step 4: Set up the integral

The volume \( V \) of the solid is given by:
\[
V = \int_{0}^{\frac{25}{16}} \pi \cdot 16y \, dy
\]
Simplify:
\[
V = 16\pi \int_{0}^{\frac{25}{16}} y \, dy
\]

This is the integral you need to evaluate to find the volume of the solid.

Would you like me to solve the integral for you or provide further details?

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**Related Questions:**
1. What is the geometric interpretation of the method of disks?
2. How would the setup change if the region were rotated about the x-axis?
3. What would the integral look like if we used the shell method instead?
4. How do you handle regions bounded by more complicated curves?
5. Can you generalize this method to rotate about any vertical line \( x = c \)?

**Tip:** Always double-check the limits of integration, especially when switching from \( x \)- to \( y \)-coordinates.

Learn how to find the volume of a solid obtained by rotating the region bounded by y = 1/16 * x^2, x = 5, y = 0 about the y-axis using the method of disks/washers. Detailed step-by-step solution provided.

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