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.