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The problem is asking to find the volume of a solid formed by rotating a region about the y-axis. The region is bounded by:

- \( y = \sqrt[4]{x} + 1 \)
- \( x = 0 \) (the y-axis)
- \( y = 2 \)

### Step-by-step Solution

#### 1. Rearranging the equation for \( y \):
Given \( y = \sqrt[4]{x} + 1 \), solve for \( x \) in terms of \( y \) to apply the method of cylindrical shells (since we are rotating around the y-axis).

Subtract 1 from both sides:
\[
y - 1 = \sqrt[4]{x}
\]

Raise both sides to the fourth power:
\[
(y - 1)^4 = x
\]

Thus, the equation becomes:
\[
x = (y - 1)^4
\]

#### 2. Volume by the Shell Method:
To find the volume using the shell method, the formula for rotating about the y-axis is:
\[
V = 2\pi \int_{y_1}^{y_2} x \cdot (f(y)) \, dy
\]
Here, \( x \) is the distance from the axis of rotation (which is just \( x \)), and \( f(y) \) is the height of the shell.

- The bounds of integration are from \( y = 1 \) (because \( y = \sqrt[4]{0} + 1 = 1 \)) to \( y = 2 \) (as given in the problem).
- The expression for \( x \) in terms of \( y \) is \( (y - 1)^4 \).

Now, compute the volume:
\[
V = 2\pi \int_{1}^{2} (y - 1)^4 \, dy
\]

#### 3. Solving the Integral:
First, expand and integrate \( (y - 1)^4 \):
\[
\int (y - 1)^4 \, dy = \frac{(y - 1)^5}{5}
\]
Evaluate this from \( y = 1 \) to \( y = 2 \):

At \( y = 2 \):
\[
\frac{(2 - 1)^5}{5} = \frac{1^5}{5} = \frac{1}{5}
\]

At \( y = 1 \):
\[
\frac{(1 - 1)^5}{5} = 0
\]

The result of the integral is:
\[
\frac{1}{5} - 0 = \frac{1}{5}
\]

#### 4. Final Volume:
Now, multiply by \( 2\pi \) to find the volume:
\[
V = 2\pi \times \frac{1}{5} = \frac{2\pi}{5}
\]

Thus, the exact volume of the solid is:
\[
V = \frac{2\pi}{5}
\]

---

Would you like more details on any step or have any specific questions?

Here are some related questions for further exploration:

1. How does the method of cylindrical shells differ from the disk method?
2. Can this volume be calculated using the disk/washer method instead?
3. What happens to the volume if the region is rotated around a different axis (e.g., \( x = 1 \))?
4. How do you determine the bounds of integration when rotating a region?
5. Can this problem be solved using parametric or polar coordinates?

**Tip:** When solving problems involving rotation, always check if cylindrical shells or disks/washers are easier to apply, based on the axis of rotation.

Find the volume of the solid that results from rotating the region bounded by the graphs of y = 4√x + 1, x = 0, and y = 2 about the y-axis. Write the exact answer.

Learn how to find the volume of the solid formed by rotating the region bounded by y = 4√x + 1, x = 0, and y = 2 about the y-axis. This problem involves using the cylindrical shell method to solve integrals for volume calculation.

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