Problem 63 - Calculus: Early Transcendentals

Question

(a) Set up an integral for the volume of a solid torus (the donut-shaped solid shown in the figure) with radii \( r \) and \( R \). (b) By interpreting the integral as an area, find the volume of the torus.

Step-by-Step Solution

Answer

The volume of the torus is \( 2\pi^2 R r^2 \).

Step 1: Understanding the Geometry of a Torus

A solid torus can be visualized as a circle of radius \( r \) revolving around a larger circle of radius \( R \). The cross-section is a circle of radius \( r \) and the central circle, around which it revolves, has radius \( R \).

Step 2: Setting Up the Integral for Volume

The volume of the torus can be considered as the volume of a circle (with radius \( r \)) revolving around a line. Using the method of disks or washers, the volume can be expressed in terms of the integral \( V = 2\pi \int_{R-r}^{R+r} (r^2 - (x-R)^2)^{1/2} (2\pi (R+x))\, dx \).

Step 3: Simplifying the Integral

We can simplify the integral by considering the contribution of the circle's revolution around the central axis: \[ V = \pi r^2 \times 2\pi R \]. This indicates that the area of the circle (\( \pi r^2 \)) is multiplied by the circumference of the path it follows (\( 2\pi R \)).

Step 4: Calculating the Volume

Compute the volume using the simplified formula: \[ V = 2\pi R \times \pi r^2 \]. Simplifying gives \( V = 2\pi^2 R r^2 \), which is the volume of the torus.

Key Concepts

Integral Calculus

Integral calculus is a fundamental branch of mathematics that involves summation of infinitesimal factors to find areas, volumes, and other related concepts. In the context of finding volumes, integrals are often used to sum up an infinite number of small pieces to get the total volume of a complex shape. When dealing with complex geometries like a torus, integrals help simplify and solve for volumes by considering infinitesimal slices. In this exercise, the problem is to determine the volume of a torus, and it's set up as an integral that accounts for the circular path of revolution around the central axis. These calculations are possible thanks to integral calculus, which allows us to sum up these infinitesimal contributions.

Geometric Visualization

Geometric visualization involves understanding and interpreting mathematical concepts through visual or spatial relationships. For a torus, imagine a small circle of radius \( r \) revolving around a larger circle with radius \( R \). This visual understanding helps in setting up the integral correctly by considering the specific path of a point on the smaller circle as it revolves around the center. Visualizing the revolutions and the resulting three-dimensional shape is crucial when determining how the integral will be set up in the formula for volume. This kind of visualization breaks down the abstract mathematical idea into concrete, understandable parts.

Revolution of Solids

The revolution of solids refers to generating a three-dimensional object by rotating a two-dimensional shape around a line, known as the axis of rotation. In this exercise, the torus is formed by revolving a circle of radius \( r \) around an axis at a distance \( R \) from the circle's center. This technique is commonly used in integral calculus to transform simple shapes like lines or curves into complex volumes. Understanding the revolution process is key to correctly setting up the integral because it defines the geometry and orientation of the solid in space, influencing the computation of its volume.

Disk Method

The disk method is a technique used in calculus to find the volume of a solid of revolution. By dividing the solid into small disks, you can apply integration to sum up the volumes of all these tiny disks. When calculating the volume of a torus using the disk method, imagine cutting the torus into many thin circular slices perpendicular to the axis of rotation. Each disk's volume can be calculated using its radius and thickness. By integrating across these slices, you sum the entire volume of the torus. This method is particularly useful because it simplifies the calculation by breaking down the complex shape into manageable units.