We are tasked with finding the volume of the solid formed when the region bounded by the curve $y = \frac{1}{\sqrt{x}}$, the line $x = \frac{1}{4}$, and the line $y = 1$, in the first quadrant, is revolved about the y-axis.

Step 1: Rearranging the curve equation

The curve is given as $y = \frac{1}{\sqrt{x}}$. To express $x$ in terms of $y$, solve for $x$:

$y = \frac{1}{\sqrt{x}} \quad \Rightarrow \quad \sqrt{x} = \frac{1}{y} \quad \Rightarrow \quad x = \frac{1}{y^2}.$

Thus, the curve can also be expressed as $x = \frac{1}{y^2}$.

Step 2: Setting up the limits

We are rotating the region bounded by the curve, the line $x = \frac{1}{4}$, and the line $y = 1$ around the y-axis. The volume of the solid can be computed using the method of cylindrical shells or the disk/washer method. Here, we will use the disk/washer method.

• The limits for $y$ are from $y = 1$ to the value of $y$ when $x = \frac{1}{4}$.

When $x = \frac{1}{4}$, substitute into $y = \frac{1}{\sqrt{x}}$:

$y = \frac{1}{\sqrt{\frac{1}{4}}} = 2.$

Thus, the limits for $y$ are from $y = 1$ to $y = 2$.

Step 3: Volume formula using the disk method

The volume $V$ of the solid of revolution is given by:

$V = \pi \int_{y_1}^{y_2} [R(y)]^2 \, dy,$

where $R(y)$ is the distance from the y-axis to the boundary of the region, which in this case is $R(y) = \frac{1}{y^2}$.

Thus, the volume integral becomes:

$V = \pi \int_{1}^{2} \left( \frac{1}{y^2} \right)^2 \, dy.$

Step 4: Simplifying and solving the integral

Simplifying $\left( \frac{1}{y^2} \right)^2 = \frac{1}{y^4}$, the volume expression becomes:

$V = \pi \int_{1}^{2} \frac{1}{y^4} \, dy.$

Now, integrate $\frac{1}{y^4}$:

$\int \frac{1}{y^4} \, dy = -\frac{1}{3y^3}.$

Now, evaluate this from $y = 1$ to $y = 2$:

= \pi \left( -\frac{1}{24} + \frac{1}{3} \right) = \pi \left( \frac{1}{3} - \frac{1}{24} \right) = \pi \cdot \frac{8}{24} = \frac{2\pi}{3}.$$ ### Final Answer: The volume of the solid is $$\frac{2\pi}{3}$$ cubic units. --- Would you like a more detailed breakdown, or do you have any further questions? Here are 5 related questions for you: 1. How does the disk method differ from the shell method for volume calculation? 2. What would change if the region were revolved about the x-axis instead? 3. How do we determine the limits of integration for a solid of revolution? 4. What would happen if the region were revolved around a different vertical line, like $$x = 1$$? 5. How would the integral change if the curve were $$y = \frac{1}{x}$$? **Tip**: When setting up volume integrals, always carefully identify the boundaries and the axis of revolution to choose the right method (disk/washer or shell).