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The Disk Method is a powerful technique used to calculate the volume of a solid of revolution. This involves creating a solid by rotating a 2D area around an axis. In this method, we visualize the solid as being composed of multiple circular disks stacked along the axis of rotation.
Each disk's diameter corresponds to a function value at a given point, and its thickness is an infinitesimally small change along the axis, typically denoted as \(dx\).
For the current problem, the area under the curve \(y = \frac{1}{x}\) from \(x = 1\) to \(x = 4\) is revolved around the x-axis, creating a series of disks.

• The radius of each disk is given by the function value \(f(x) = \frac{1}{x}\).
• The volume of a single disk is calculated by \(( \pi (\text{radius})^2 \cdot \text{thickness} )\) which corresponds to \(\pi \left( \frac{1}{x} \right)^2 dx\).

By summing up the volumes of all these infinitesimally thin disks, we find the total volume of the solid. This sum is expressed as the integral \(V = \pi \int_{1}^{4} \left( \frac{1}{x} \right)^2 dx\).

Integration is the core process of calculus used for finding the area under a curve or the accumulation of quantities. In the context of volume of revolution, integration helps us add up the infinitesimally small volumes of disks along the axis of rotation.
For this exercise, the function given is \(f(x) = \frac{1}{x}\), and by applying the disk method, we set up the integral \(\pi \int_{1}^{4} \left( \frac{1}{x} \right)^2 dx\).
To solve the integral \(\int \frac{1}{x^2} dx\), we use the power rule which states that the integral of \(x^n\) (where \ is not equal to -1) is \((1/(n+1))x^{n+1} + C\).

• Applying the power rule, \(\int x^{-2} dx = \frac{-1}{x} + C\).
• We evaluate this at the limits \(x = 1\) and \(x = 4\) to find the definite integral, which checks out to be \(\frac{3}{4}\).

This result is then used to find the total volume, confirming our calculations.

Approaching calculus problems, particularly those involving the volume of revolution, requires a step-by-step problem-solving strategy that helps break down complex problems into manageable parts.
Here's a simple guide to tackle such problems:

• **Visualize the Problem:** Sketch the graph of the function to accurately understand the region being rotated. This helps in setting up the correct limits of integration.
• **Set Up the Integral:** Choose the appropriate method (disk, washer, or shell) to set up the integral. For this exercise, the Disk Method is chosen, leading to the correct integral formula.
• **Solve the Integral:** Use calculus techniques like the power rule or substitution to solve the definite integral. Integrate the function over the given interval and apply the limits correctly.
• **Check Your Work:** Double-check the integration and calculation steps. Reviewing ensures clarity and prevents mistakes.

Being methodical not only helps achieve the correct answer but also builds a deeper understanding of the concepts involved. Practice and familiarity with these steps make calculus problems more approachable.