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Chapter 6: Problem 41

Use the Theorem of Pappus and the fact that the area of an ellipse with semiaxes \(a\) and \(b\) is \(\pi a b\) to find the volume of the elliptical torus generated by revolving the ellipse $$ \frac{(x-k)^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $$ about the \(y\) -axis. Assume that \(k>a\).

Short Answer

Expert verified
The volume of the elliptical torus is \(2\pi^2 k ab\).

Step by step solution

01

Identify the Ellipse and Its Properties

The given ellipse is \( \frac{(x-k)^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \). The center of the ellipse is at \( (k, 0) \) and its semiaxes are \( a \) (horizontal) and \( b \) (vertical). The ellipse will revolve around the \( y \)-axis to form a torus.
02

Calculate the Path of the Center Rotation

When the ellipse is revolved around the \( y \)-axis, the path of the center of the ellipse is a circle with a radius of \( k \). This circle is perpendicular to the \( y \)-axis and lies in the \( xy \)-plane.
03

Use Pappus鈥檚 Second Theorem

The Theorem of Pappus states that the volume \( V \) of a solid of revolution generated by rotating a plane area \( A \) about an external axis is given by: \[ V = 2 \pi d \times A \] where \( d \) is the distance traveled by the centroid of the area \( A \).
04

Determine the Area of the Ellipse

The area \( A \) of an ellipse with semiaxes \( a \) and \( b \) is \( \pi a b \). This will be the area rotating around the \( y \)-axis.
05

Determine the Volume of the Torus

Substituting the area \( A = \pi ab \) and the path length \( d = 2\pi k \) into Pappus's theorem: \[ V = 2 \pi k \times \pi ab = 2 \pi^2 k ab. \] Therefore, the volume of the elliptical torus is \( 2 \pi^2 k ab \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pappus's Theorem
The theorem of Pappus is a fundamental concept in geometry used to calculate volumes of solids of revolution. It focuses on determining the volume generated by rotating a two-dimensional shape around an external axis.

**Key Points to Understand:**
  • Pappus's Second Theorem states that the volume \(V\) of a solid of revolution formed by rotating a plane area \(A\) around an axis outside the shape is given by \(V = 2 \pi d \times A\),
  • \(d\) is the distance traveled by the centroid of the shape, which is a point that marks the center of mass of the area.
  • The theorem simplifies the often complex calculations involved in finding the volume of intricate shapes like tori or complex solids obtained by revolution.
In the given exercise, Pappus's Theorem helps in calculating the volumetric size of the torus formed by revolving an ellipse around the y-axis. The centroid's circular path plays a crucial role in this computation.
Ellipse Area
An ellipse is a symmetrical shape that looks like a stretched circle. When dealing with ellipses, it is important to understand their area.

**Understanding the Area of an Ellipse:**
  • An ellipse is defined by its semiaxes, \(a\) and \(b\), which are half the respective major and minor axes.
  • The formula for the area of an ellipse is \(\pi ab\), where \(a\) is the horizontal semiaxis, and \(b\) is the vertical semiaxis.
The area is a crucial part of using Pappus's theorem since the calculated area determines the volume of the torus formed during rotation. It's important to identify these semiaxes correctly to apply the formula accurately. For the given ellipse, the semiaxes are aligned horizontally and vertically, making it straightforward to apply this concept.
Solid of Revolution
A solid of revolution is created by rotating a two-dimensional shape around an axis. This concept is essential for understanding how an elliptical torus is formed by rotating an ellipse around an axis.

**Essential Characteristics:**
  • Solids of revolution can include a variety of shapes like spheres, cylinders, tori, etc., derived by rotating different geometric figures.
  • The axis of rotation is critical. It could be inside the shape (like a circle forming a sphere) or external (like a circle creating a torus).
  • Pappus's Theorem often aids in determining the volume of such solids by calculating the area that rotates and the path length, thereby simplifying complex volumetric calculations.
In this particular case, the exercise involves forming a torus by revolving an ellipse around the y-axis, resulting in a distinctive donut-like shape with calculable volume.
Torus
A torus is a doughnut-shaped solid often used to represent solids of revolution in mathematics. It provides an interesting example of how different geometric and algebraic concepts come together.

**Characteristics of a Torus:**
  • A torus can be formed by revolving a circle or an ellipse around a line exterior to the shape, usually resulting in a ring-like structure.
  • The central idea in forming a torus is the path of the shape's center during revolution, which follows a circular trajectory, marking the torus's larger circumference.
  • The volume of a torus can be computed with precision using the area of the initial shape that revolves and the distance the centroid travels, often resulting in a clean application of Pappus's theorem.
In the exercise, by revolving an ellipse about the y-axis, we form an elliptical torus, which illustrates the complex yet fascinating nature of solids of revolution and how theoretical concepts are applied to yield tangible metrics like volume.

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