91Ó°ÊÓ

The disk method allows us to find the volume of a solid of revolution by integrating cross-sectional areas perpendicular to the axis of rotation. Imagine slicing the solid into thin disks, each with a small thickness \(\text{dx}\).
To find the volume of each of these disks, we use the formula for the volume of a cylinder: \[ V_{\text{disk}} = \text{area of base} \times \text{height} \]Here, the 'base' is a circular cross-section, and the 'height' is the thickness \(\text{dx}\).
If the radius of each disk is given by \( R(x) \), the volume of a thin disk becomes: \[ V_{\text{disk}} = \pi (R(x))^2 \text{dx} \]To find the total volume, integrate this expression over the interval of interest: \[ V = \int_{a}^{b} \pi [R(x)]^2 dx \]

In the context of this exercise, a solid of revolution is formed by rotating a region bounded by graphs around an axis. When you rotate a curve like \( y = \frac{1}{x} \) around the x-axis, it creates a 3D shape.
To visualize it, think of tracing the area under the curve between \( x = 1 \) and \( x = 3 \). When this area rotates, it sweeps out a solid shape.
The total volume of this shape can be found using the disk method. Each disk has a radius equal to the value of the function at that point, \( y = \frac{1}{x} \), so the radius is \( R(x) = \frac{1}{x} \). Plug this into the disk method formula and integrate over the given interval.

Finding the volume of a solid of revolution is an exciting application of integral calculus. It shows how powerful integration can be in solving real-world problems.
The steps are as follows:

• Identify the region to be rotated.
• Determine the axis of rotation and establish the function's formula.
• Use the disk method formula: \[ V = \pi \int [R(x)]^2 dx \]
• Integrate over the specified bounds.

The principles you learn here apply to many other problems in calculus, including physics and engineering. Whether calculating the volume of a tank or the flow of a liquid, these techniques provide the foundation for understanding and solving these challenges.