Let's solve this problem of finding the volume of the solid obtained by rotating the region bounded by the curves $y = \frac{1}{x^4}$, $y = 0$, $x = 2$, and $x = 9$ around the y-axis, using the shell method, washer method, and disk method (where applicable).

Step 1: Shell Method

The shell method involves slicing the region into cylindrical shells, and integrating along the axis of rotation. The formula for the volume using the shell method is:

$V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx$

For the given curves, we use $y = \frac{1}{x^4}$ as the function, and rotate about the y-axis. The radius of each shell is $x$, and the height of the shell is $f(x) = \frac{1}{x^4}$.

The limits for $x$ are from 2 to 9.

$V = 2\pi \int_{2}^{9} x \cdot \frac{1}{x^4} \, dx = 2\pi \int_{2}^{9} \frac{1}{x^3} \, dx$

Now, solve the integral:

$V = 2\pi \int_{2}^{9} x^{-3} \, dx = 2\pi \left[ \frac{x^{-2}}{-2} \right]_{2}^{9}$

$V = 2\pi \left[ \frac{-1}{2x^2} \right]_{2}^{9}$

$V = 2\pi \left( \frac{-1}{2 \cdot 9^2} - \frac{-1}{2 \cdot 2^2} \right)$

$V = 2\pi \left( \frac{-1}{162} + \frac{1}{8} \right)$

$V = 2\pi \left( \frac{-1}{162} + \frac{21}{168} \right)$

$V = 2\pi \cdot \frac{41}{168}$

$V = \frac{41\pi}{84}$

Thus, the volume using the shell method is:

$V = \frac{41\pi}{84}$

Step 2: Washer Method

The washer method requires slicing the solid perpendicular to the axis of rotation (in this case, the y-axis). However, the washer method is generally better suited for regions rotated about a horizontal axis, such as the x-axis, and doesn't apply directly for a rotation about the y-axis without significant changes to the method.

For this specific problem, the washer method is not the optimal choice for rotation about the y-axis. Instead, we use either the shell method (as done) or disk method for appropriate rotations.

Step 3: Disk Method

The disk method is also typically used for rotations around the x-axis or other horizontal axes. For a rotation around the y-axis, the disk method is not easily applicable because we would need to express $x$ as a function of $y$, and this would become cumbersome for $y = \frac{1}{x^4}$.

Therefore, the disk method is not useful in this case.

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Final Answer

The volume of the solid obtained by rotating the region bounded by $y = \frac{1}{x^4}$, $y = 0$, $x = 2$, and $x = 9$ around the y-axis is:

$V = \frac{41\pi}{84}$

Would you like any more details on this process or further clarifications?

Related Questions:

1. How does the method of cylindrical shells compare to the disk/washer method for this type of problem?
2. Can you provide more examples where the washer method would be useful?
3. What happens if we rotate around a different axis, like the x-axis?
4. How would the problem change if the region was unbounded on one side (e.g., $x \to \infty$)?
5. Could we use numerical methods to approximate the volume in more complex cases?

Tip:

The shell method is often the best choice when rotating around the y-axis, as it simplifies the calculation of cylindrical shells formed parallel to the axis of rotation.