15. Find the volume of the solid generated by revolving the region bounded by y = 2*(rootx)
y=2, x=4, about the line y=2.
x and the lines

To find the volume of the solid generated by revolving the region bounded by \( y = 2\sqrt{x} \), \( y = 2 \), \( x = 4 \), and \( x = 0 \) (since the region is bounded from \( x = 0 \) to \( x = 4 \)) about the line \( y = 2 \), we use the **washer method**.

### Step 1: Identify the curves and boundaries.
- The curve is \( y = 2\sqrt{x} \).
- The region is bounded by \( y = 2 \) (the horizontal line), \( x = 4 \) (a vertical line), and \( x = 0 \) (the y-axis).

### Step 2: Set up the washer method integral.
The volume \( V \) is given by:
\[
V = \pi \int_{a}^{b} \left[\left(\text{outer radius}\right)^2 - \left(\text{inner radius}\right)^2\right] dx
\]
Here:
- The outer radius is the distance from the line \( y = 2 \) to the \( x \)-axis (which is 0, since \( y=2 \) is being revolved around).
- The inner radius is the distance from the curve \( y = 2\sqrt{x} \) to \( y = 2 \).

### Step 3: Express the radii.
- Outer radius \( R(x) = y_{\text{upper}} - y_{\text{axis}} = 2 - 2 = 0 \)
- Inner radius \( r(x) = y_{\text{upper}} - y_{\text{lower}} = 2 - 2\sqrt{x} \)

### Step 4: Set up and compute the integral.
\[
V = \pi \int_{0}^{4} \left[ \left(2 - 0\right)^2 - \left(2 - 2\sqrt{x}\right)^2 \right] dx
\]
Simplifying the expression inside the integral:
\[
V = \pi \int_{0}^{4} \left[4 - (4 - 8\sqrt{x} + 4x)\right] dx
\]
\[
V = \pi \int_{0}^{4} \left(8\sqrt{x} - 4x\right) dx
\]

### Step 5: Integrate.
\[
V = \pi \left[\frac{8 \cdot x^{3/2}}{3/2} - \frac{4x^2}{2}\right]_{0}^{4}
\]
\[
V = \pi \left[\frac{16x^{3/2}}{3} - 2x^2\right]_{0}^{4}
\]
Evaluating the integral:
\[
V = \pi \left[\frac{16 \cdot 8}{3} - 2 \cdot 16\right]
\]
\[
V = \pi \left(\frac{128}{3} - 32\right) = \pi \left(\frac{128 - 96}{3}\right) = \pi \left(\frac{32}{3}\right)
\]

### Final Answer:
\[
V = \frac{32\pi}{3}
\]

So, the volume of the solid is \( \frac{32\pi}{3} \) cubic units.

Would you like a detailed explanation of any step or have any questions?

### Related Questions:
1. How would the method change if the solid was revolved around the x-axis instead?
2. What is the significance of the "washer method" versus the "disk method"?
3. How would you approach the problem if the curve was revolved around \( y = -2 \) instead?
4. Can you generalize this method to other types of functions?
5. What if the region was bounded by a different curve, say \( y = x^2 \), how would you set up the integral?

### Tip:
When using the washer method, always clearly distinguish between the inner and outer radii to avoid mistakes in setting up your integral.

Learn how to find the volume of the solid generated by revolving the region bounded by y = 2*sqrt(x), y = 2, x = 4 about the line y = 2 using the washer method. Includes step-by-step integral setup and computation.

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