To find the volume of the solid of revolution obtained by rotating the given region about the $y$-axis, we use the method of cylindrical shells or the washer method. Here, the washer method is appropriate.

Step 1: Analyze the bounds

• The region is bounded by:
  1. $y = \sqrt{x}$
  2. $y = \frac{1}{x}$
  3. $x = 4$

We rewrite these in terms of $x$ to express $x$ as a function of $y$:

• $y = \sqrt{x} \implies x = y^2$
• $y = \frac{1}{x} \implies x = \frac{1}{y}$

For $y$-bounds:

• At $x = 4$, $y = \sqrt{4} = 2$ for $y = \sqrt{x}$, and $y = \frac{1}{4}$ for $y = \frac{1}{x}$.
• So, the bounds of integration are $y \in \left[\frac{1}{4}, 2\right]$.

Step 2: Set up the volume integral

The volume is given by the formula for solids of revolution about the $y$-axis:

$V = \pi \int_{y_{\text{min}}}^{y_{\text{max}}} \left[\text{Outer radius}^2 - \text{Inner radius}^2\right] \, dy$

• The outer radius is given by $x = 4$ (fixed vertical line).
• The inner radius is the $x$-coordinate of the curve $y = \frac{1}{x}$, rewritten as $x = \frac{1}{y}$.

Thus, the volume integral becomes: $V = \pi \int_{\frac{1}{4}}^{2} \left[4^2 - \left(\frac{1}{y}\right)^2\right] \, dy$

Step 3: Simplify and compute the integral

$V = \pi \int_{\frac{1}{4}}^{2} \left[16 - \frac{1}{y^2}\right] \, dy$

Separate the integral: $V = \pi \left[ \int_{\frac{1}{4}}^{2} 16 \, dy - \int_{\frac{1}{4}}^{2} \frac{1}{y^2} \, dy \right]$

First integral:

$\int_{\frac{1}{4}}^{2} 16 \, dy = 16 \left[y\right]_{\frac{1}{4}}^{2} = 16 \left(2 - \frac{1}{4}\right) = 16 \cdot \frac{7}{4} = 28$

Second integral:

$\int_{\frac{1}{4}}^{2} \frac{1}{y^2} \, dy = \left[-\frac{1}{y}\right]_{\frac{1}{4}}^{2} = -\frac{1}{2} - \left(-4\right) = -\frac{1}{2} + 4 = \frac{7}{2}$

Step 4: Combine results

$V = \pi \left[28 - \frac{7}{2}\right] = \pi \cdot \frac{56 - 7}{2} = \pi \cdot \frac{49}{2} = \frac{49\pi}{2}$

Final Answer:

$\boxed{\frac{49\pi}{2}}$

Would you like further details or another example? Here are 5 related questions you might find helpful:

1. How do we find the volume of solids using the disk method?
2. How does the cylindrical shell method compare to the washer method for this problem?
3. Can this integral be solved numerically instead of analytically?
4. How does the volume change if the rotation is around the $x$-axis?
5. What happens if the boundary $x = 4$ is replaced with another value?

Tip: Always visualize the region and axis of rotation to ensure correct setup of integrals.