Problem 51 - Thomas Calculus in SI Units

Question

The volume of a torus The disk \(x^{2}+y^{2} \leq a^{2}\) is revolved about the line \(x=b(b>a)\) to generate a solid shaped like a doughnut and called a torus. Find its volume. (Hint: \(\int_{-a}^{a} \sqrt{a^{2}-y^{2}} d y=\pi a^{2} / 2,\) since it is the area of a semicircle of radius \(a .)\)

Step-by-Step Solution

Answer

The volume of the torus is \(2 \pi^2 a^2 b\).

Step 1: Understand the Problem

We need to find the volume of a torus formed by revolving a disk described by the inequality \(x^2 + y^2 \leq a^2\) about a vertical line at \(x=b\) where \(b > a\). Visualize this as a circle with radius \(a\) revolving at a distance \(b\) from its center.

Step 2: Set up the Formula for Volume

The volume \(V\) of revolution about a vertical line can be calculated using the formula for volume of solid of revolution:\[ V = 2\pi \int_{c}^{d} x f(x) dx \]However, as we are revolving around \(x=b\), adjusting for cylindrical shells, the volume for a small element is \[ dV = 2\pi(b-x)\sqrt{a^2-y^2} dy \].

Key Concepts

Solid of Revolution

A solid of revolution is a three-dimensional object created by rotating a two-dimensional shape around a straight line, typically known as the axis of rotation. In our problem, the shape being revolved is a disk, specifically defined by the equation \(x^2 + y^2 \leq a^2\), and we revolve this disk about a line that is a distance \(b\) away from its center. This line, in our case, is specified as \(x = b\), where \(b > a\). The resulting solid, known as a torus, resembles a doughnut.
Understanding the concept of a solid of revolution is key to approaching this problem, as it informs us how to apply appropriate mathematical techniques to calculate the volume. It’s helpful to visualize the process: imagine the disk spinning around the given vertical line, sweeping out a three-dimensional space that forms the torus. Such visualizations anchor the abstract concept in concrete reality, making it easier to grasp how the disk transforms into a torus.

Cylindrical Shell Method

The cylindrical shell method is especially useful in calculating the volume of solids of revolution when the axis of rotation is parallel to the axis of integration. This method involves conceptualizing the solid as being composed of 'cylindrical shells'. Each shell corresponds to a thin vertical strip of the graph, revolved around the axis.
For the problem at hand, each small vertical strip at a distance \(b-x\) from the line \(x = b\) creates a cylindrical shell when rotated. The formula used for a cylindrical shell is given by:

• Height of the shell: \(\sqrt{a^2-y^2}\)
• Radius of the shell: \(b-x\)
• Thickness of the shell: \(dy\)

Thus, the volume of each shell is \(dV = 2\pi(b-x)\sqrt{a^2-y^2} dy\). By calculating the volume for each cylindrical shell and integrating, we find the total volume of the torus. This approach is intuitive for problems like ours, where revolving shapes are not aligned with standard axes.

Volume Integration

Volume integration is a mathematical process of summing up small volumes to find the total volume of a solid. In our exercise, integration is used to sum the volumes of infinitely many thin cylindrical shells to ascertain the full volume of the torus.
The integration captures the cumulative effect of all these thin cylindrical slices swept around our axis of revolution. We integrate with respect to \(y\), from \(-a\) to \(a\). This accounts for the entire disk, extending vertically from the lowest to the highest points of the disk. With the formula \(dV = 2\pi(b-x)\sqrt{a^2-y^2} dy\), setting the integration limits from \(-a\) to \(a\), captures each thin ring’s contribution and sums them to compute the volume.
It is fascinating to observe how the abstract concept of integration, often difficult to visualize, translates into finding the tangible measurements of solids like the torus. This enhancer further understanding of how mathematics models the physical world.

Calculus Problem Solving

Calculus problem-solving often involves a step-by-step approach, starting with understanding the problem and mathematical concepts involved, then formulating the equations needed to solve it. Dealing with the volume of a torus showcases this methodology well.
First, we clearly understand the problem and visualize the revolved solid. Then, identifying the appropriate method—here the cylindrical shell method—is critical. The creative application of this method involves breaking down the solid into manageable elements and setting up integrals that describe each element's contribution to the whole.
Finally, executing the integral correctly involves meticulous calculations and often creative mathematical manipulation. Each of these steps emphasizes the organized approach calculus offers for meticulous problem-solving, balancing both algebraic and geometric reasoning to provide real-world solutions. Practicing such problems enhances not only mathematical intuition but also broader analytical skills applicable in various disciplines.