Problem 38 - Calculus of a Single Variable

Question

Use the Theorem of Pappus to find the volume of the solid of revolution. The torus formed by revolving the circle \(x^{2}+(y-3)^{2}=4\) about the \(x\) -axis

Step-by-Step Solution

Answer

The volume of the solid of revolution, or the torus, is \(24π^{2}\).

Step 1: Identify the radius of the circle and its centre

From the equation \(x^{2} + (y-3)^{2} = 4\), we can see that the center of the circle is (0,3) and the radius r is 2.

Step 2: Determine the distance to the centroid'

The distance from the x-axis to the centroid of the circle is the distance from the x-axis to the center of the circle, which is 3 units. Therefore, k = 3.

Step 3: Determine the area of the circular region

The area A of the circular region is given by A = πr^{2}. Plugging r=2 into this equation gives A = π(2)^{2} = 4π.

Step 4: Apply the Theorem of Pappus

Now we apply the Theorem of Pappus: V = (2πk)A. Substitute k=3 and A=4π, we get V = 2π(3)(4π) = 24π^{2}.

Key Concepts

Theorem of Pappus

The Theorem of Pappus provides a powerful technique to find the volume of a solid of revolution without intensive calculus. When a plane figure is revolved around an external axis, the volume of the resulting solid can be calculated easily. The theorem states that the volume is the product of the area of the figure, the path traced by the centroid, and the distance it travels. This path distance is essentially the circumference of the circle formed by the centroid as it revolves. Therefore, you need two main calculations: the area of the cross-section and the path of the centroid. In mathematical terms, this is expressed as \( V = (2\pi k)A \), where \( k \) is the distance from the axis to the centroid, and \( A \) is the area of the shape revolved.

torus volume calculation

A torus is a donut-shaped object that results from revolving a circle in three-dimensional space. By using the Theorem of Pappus, we can efficiently calculate its volume. The torus forms by revolving a circle about an axis that lies in the same plane but does not intersect the circle. To find its volume:

• Identify the circle's radius and find its area \( A = \pi r^2 \).
• Determine the distance (\( k \)) from the centroid of the circle to the axis of revolution.
• Apply the formula \( V = (2\pi k)A \) to get the total volume.

For our exercise, a circle with radius 2 and area \(4\pi\) was revolved about the x-axis. The centroid distance \( k \) was 3. Thus, the torus volume was calculated as \( 24\pi^2 \). Employing the Theorem of Pappus simplifies such computations, especially compared to directly integrating in three dimensions.

circle geometry

Understanding circle geometry is critical here. The circle in question is defined by the equation \( x^2 + (y-3)^2 = 4 \). This equation tells us:

• The center is at \((0, 3)\).
• The radius is 2 because the equation is in the standard form \((x-h)^2 + (y-k)^2 = r^2\).

These simple circle properties are the foundation for determining the area and centroid distance, both crucial for applying the Theorem of Pappus. With the radius, you can swiftly calculate the area of the circular cross-section, and the circle's center helps determine the centroid's position relative to the x-axis.

revolution about x-axis

Revolution about the x-axis involves rotating a shape around this horizontal axis. The process creates a three-dimensional object from a two-dimensional shape. For revolutions, the x-axis serves as the line of revolution, and the entire figure traces out a volume in space.

• The figure's centroid is pivotal since its path defines the solid's size.
• The distance from the centroid to the axis (\( k \)) greatly influences the resulting volume, as seen in calculating the torus volume.

In essence, by revolving a circle with a known equation, we leverage its geometry to grasp how the shape impacts the space. Such a revolution transforms simple geometric properties into tools for solving complex volumetric problems. The interplay between two-dimensional geometry and three-dimensional space is how we bridge into solid geometry using revolutions.