Question

Use Pappus's Theorem to find the volume of the torus obtained when the region inside the circle \(x^{2}+y^{2}=a^{2}\) is revolved about the line \(x=2 a\)

Short answer

Volume of the torus is \(4\pi^2 a^3\).

Step-by-step solution

Identify the Geometric Parameters

We start by identifying the geometric parameters given. The circle is defined by the equation \(x^2 + y^2 = a^2\) which has a radius of \(a\) and is centered at the origin (0,0). This circle is revolved around the vertical line \(x = 2a\), an axis that is \(2a\) units away from the circle's center.

Understand Pappus’s Centroid Theorem for Volume

Pappus’s Centroid Theorem states that the volume \(V\) of a solid of revolution generated by rotating a plane figure \(F\) about an axis external to \(F\) is equal to the product of the area of \(F\) and the distance traveled by the centroid of \(F\). The formula is \ V = A \, (2\pi d)\, where \(A\) is the area of the region, and \(d\) is the distance from the centroid of the region to the axis of rotation.

Calculate the Area of the Circle

The area \(A\) of a circle with radius \(a\) is calculated using the formula \(A = \pi a^2\). This represents the area of the region \(F\) that will be revolved.

Determine the Centroid's Distance to the Axis

Since the circle is symmetric about its center, the centroid of the circular region is at the origin (0,0). The distance \(d\) from the center of the circle (centroid) to the axis \(x = 2a\) is simply \(2a\).

Apply Pappus’s Theorem to Find the Volume

Apply Pappus's Theorem: substitute the values calculated into the theorem formula \(V = A \, (2\pi d)\). Here, \(A = \pi a^2\) and \(d = 2a\). Thus the volume \(V = \pi a^2 \, (2 \pi \times 2a) = 4\pi^2 a^3\). This expresses the volume of the torus.

Key concepts

Pappus's Theorem

Pappus's Theorem, also known as Pappus's Centroid Theorem, plays a significant role in geometry, particularly when dealing with the volumes of solids formed by rotation, known as solids of revolution.
This theorem provides a simple way to find the volume of such solids without delving into complicated integration. In essence, it states that the volume of a solid of revolution is the product of the area of the rotating shape and the circumference of the path traveled by its centroid during the rotation. This can be written as:

• \[ V = A imes (2 \pi d) \]

Where:

• \( V \) is the volume of the solid
• \( A \) is the area of the plane figure being rotated
• \( d \) is the distance the centroid travels as it revolves around the axis

Using Pappus's Theorem simplifies the process, as long as you know the figure's area and the centroid's path length. This becomes especially useful when dealing with common geometric shapes like circles.
From this theorem, the calculation becomes straightforward and highlights the genius of Pappus, who found a geometric way to conceptualize volume through movement and space.

Solid of Revolution

A solid of revolution is formed when a two-dimensional shape, such as a circle or rectangle, is rotated around an axis. This creates a symmetrical three-dimensional object.
The concept is pivotal in calculating volumes of objects that might not be straightforward to measure using basic geometry. Imagine having a shape like a semi-circle; when revolved around the base, it turns into a sphere. By using this method, several complex volumes become easy to calculate.
The key to understanding solids of revolution is recognizing how the original shape transforms as it spins around the axis. The axis can be inside, outside, or along the edge of the shape, and the position of the axis will greatly influence the resulting solid.
In many calculus problems, the axis is usually along one of the coordinate axes, but it can also be parallel or at a specific distance, as seen in various exercises. Learning to visualize how shapes transform gives a robust understanding of real-world objects formed by rotational symmetry, making this concept invaluable in engineering and physics.

Torus Volume Calculation

Calculating the volume of a torus involves understanding both circular geometry and principles of solids of revolution.
A torus resembles the shape of a doughnut and is created by revolving a circle around an external axis that is parallel to the circle. Using the problem as an example, we have a circle with equation \(x^2 + y^2 = a^2\), and it revolves around the line \(x = 2a\). This external axis is at a distance from the circle's center producing the doughnut-like shape.
To find the volume, we cleverly use Pappus's Theorem. First, determine the area \(A\) of the circle, which is \(\pi a^2\). The path traveled by the centroid, which is the center of the circle, has a distance of \(2\pi \times 2a\) once it revolves fully, giving it a circular motion factor.
Therefore, the volume \(V\) of the torus is given by:

• \[ V = \pi a^2 \times (2\pi \times 2a) \]

Simplifying this leads to:

• \[ V = 4\pi^2 a^3 \]

This provides the formula for calculating the volume of a torus, illustrating how geometry and calculus effectively converge to solve complex real-world problems.