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 lines.

*x* = *y*2,   *y* = *x*/4 , around the line y=-4

    


To find the volume \( V \) of the solid obtained by rotating the region bounded by the curves \( x = y^2 \) and \( y = \frac{x}{4} \) around the line \( y = -4 \), we use the method of washers.

### Step 1: Find the points of intersection
First, determine the points of intersection between the two curves by setting \( x = y^2 \) equal to \( x = 4y \):

\[
y^2 = \frac{x}{4} \quad \text{and} \quad x = y^2
\]

Substitute \( x = 4y \) into \( y^2 = \frac{x}{4} \):
\[
y^2 = \frac{4y}{4} = y
\]
\[
y(y - 1) = 0
\]

Thus, \( y = 0 \) or \( y = 1 \). 

For \( y = 0 \), \( x = 0^2 = 0 \), and for \( y = 1 \), \( x = 1^2 = 1 \), so the curves intersect at \( (0,0) \) and \( (1,1) \).

### Step 2: Express the radii for the washer method
The curves are rotated about the line \( y = -4 \). The radius of a washer is determined by the distance from each curve to the line \( y = -4 \).

- **Outer radius (R(y))**: The distance from the line \( y = -4 \) to the curve \( y = \frac{x}{4} \). Since \( x = 4y \), we rewrite it as:
  \[
  y = \frac{x}{4} \quad \text{so} \quad R(y) = \left(\frac{x}{4} + 4\right)
  \]

  Substituting \( x = 4y \), we get:
  \[
  R(y) = y + 4
  \]

- **Inner radius (r(y))**: The distance from the line \( y = -4 \) to the curve \( x = y^2 \). We solve for \( y \) from \( x = y^2 \), which gives:
  \[
  r(y) = y^2 + 4
  \]

### Step 3: Set up the integral

The volume of the solid is given by the integral:
\[
V = \pi \int_{0}^{1} \left[ (R(y))^2 - (r(y))^2 \right] dy
\]

Substitute the expressions for \( R(y) \) and \( r(y) \) into the integral:
\[
V = \pi \int_{0}^{1} \left[ (y + 4)^2 - (y^2 + 4)^2 \right] dy
\]

### Step 4: Simplify and evaluate the integral

Expand the squares inside the integrand:
\[
V = \pi \int_{0}^{1} \left[ (y^2 + 8y + 16) - (y^4 + 8y^2 + 16) \right] dy
\]

Simplify the integrand:
\[
V = \pi \int_{0}^{1} \left[ -y^4 - 7y^2 + 8y \right] dy
\]

This integral can be computed to find the exact volume of the solid.

Would you like to go through the solution of this integral or need any more details?

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**Related Questions:**
1. How would the setup change if the region were rotated about the x-axis?
2. What is the geometric interpretation of the method of washers?
3. How do you handle cases where the curves are more complex functions?
4. What if the rotation axis was above the region, say \( y = 4 \)?
5. Can this method be extended to polar coordinates?

**Tip:** When rotating around lines other than the coordinate axes, always remember to adjust the radii by adding or subtracting the distance to the line of rotation.

Learn how to find the volume of a solid obtained by rotating the region bounded by the curves x = y^2 and y = x/4 around the line y = -4 using the method of washers. Includes step-by-step integral setup and solution.

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