91Ó°ÊÓ

The disk method is a technique used in calculus to find the volume of a solid of revolution. This method is particularly effective when a region is being rotated around an axis. To visualize, imagine cutting the solid into very thin disks perpendicular to the axis of rotation.
Each disk can be considered as a thin cylinder. The volume of each is approximately \(\pi (r^2)h\), where \(r\) is the radius of the disk, and \(h\) is its thickness.

• The **radius** of each disk corresponds to the distance from the curve to the axis of rotation.
• The **thickness** is an infinitesimally small change along the axis of rotation, denoted by \(dy\) or \(dx\).

For this exercise, we use a simple form of the disk method as there is no inner radius (another curve closer to the axis). This simplifies our setup to a single integral representing the total volume.

Definite integration is a fundamental concept in calculus used to find exact values of integrals, particularly useful when calculating the total area under a curve between two bounds. In our case, it is crucial for determining the volume of a solid.
The notation for definite integration is \(\int_{a}^{b} f(y) \,dy\), where \(a\) and \(b\) are the limits of integration, reflecting the range over which the function \(f(x)\) is integrated.

• **Boundaries**: They are essential as they specify the segment of the curve being evaluated.
• **Evaluation**: Once we have the antiderivative, we substitute these boundaries to find the value.

In the solution, integration was performed from the lower boundary \(y = 1\) to the upper boundary \(y = 2\). The process included integrating the function \(y^2 - 1\), resulting in a simplified expression for the volume. The ultimate step involved precise evaluation to conclude the definite integral.

A hyperbola is a type of conic section characterized by curves that resemble two opposite-facing parabolas. The standard form of a hyperbola's equation used in this problem is \(y^2 - x^2 = 1\).
Understanding hyperbolas involves recognizing the general behavior of these curves:

• The **asymptotes**: Hyperbolas have asymptotes which the branches tend to but never touch.
• The **center** and **vertices**: The center is situated where the axes intersect, and the vertices are the closest points on each branch to the center.

In this specific exercise, the hyperbola is central, forming a critical boundary. Its integration contributes to shaping the defined volume when revolved around the x-axis.

Rotation about the x-axis refers to spinning a region in the xy-plane around the x-axis, creating a solid of revolution.
This method of rotation is common when considering functions bounded by certain limits or constraints:

• **Visualization**: Picture a region in 2D and imagine trailing it along a path around the x-axis.
• **Application**: Often applies to various geometric shapes and curves, including parabolas, circles, and hyperbolas.

For the provided exercise, the region formed by part of a hyperbola and a horizontal line is revolved around the x-axis, resulting in a symmetrical 3D shape. Understanding this rotation is pivotal to establishing the context for the disk method and the respective volume calculation.