Problem 43 - Essential Calculus

Question

{Volume of a Torus } A torus is formed by revolving the region bounded by the circle \(x^{2}+y^{2}=1\) about the line \(x=2\) (see figure). Find the volume of this "doughnut-shaped" solid. (Hint: The integral \(\int_{-1}^{1} \sqrt{1-x^{2}} d x\) represents the area of a semicircle.)

Step-by-Step Solution

Answer

22\frac{1}{3}\pi cubic units

Step 1: Understand the Disk Method to calculate Volume

The formula for finding the volume of the solid of revolution using the Disk Method is given by \[ V = \pi \int_a^b [R(x)]^{2} dx \] where R(x) represents the distance from the axis of revolution to the furthest edge of the region being revolved.

Step 2: Calculate the Radius

The circle's equation is \(x^{2}+y^{2}=1\).This can be written as \(y=\sqrt{1-x^{2}}\) for the semicircle above the x-axis.The radius (R(x)) of each disk is represented by the distance from the line \(x = 2\) to any point on the semicircle.So, \( R(x)=2-x \) for \(-1 \leq x \leq 1\).

Step 3: Substitute radius in the Volume formula and compute the integral

Substitute the radius \( R(x)=2-x \) in the Disk Method formula and solve the integral. \[ V = \pi \int_{-1}^{1} [(2-x)]^{2} dx \] = \(\pi [4x-\frac{x^3}{3}]_{-1}^{1}\) = \(\pi \left[ \left(4(1)-\frac{1^3}{3}\right) - \left(4(-1)-\frac{(-1)^3}{3}\right) \right]\) = \(\pi[8-\frac{2}{3}]\)

Step 4: Final Calculation

After calculating, get the final volume of the Torus: Volume = \(22\frac{1}{3}\pi\) cubic units.

Key Concepts

Disk Method

The Disk Method is an essential technique in calculus for finding the volume of a solid of revolution. It involves slicing the solid into thin, disk-shaped sections whose volumes can be summed to find the total volume. This method is particularly useful when the solid has a uniform thickness throughout its structure, which is perpendicular to the axis of revolution.

The formula for the Disk Method is given by:\[V = \pi \int_a^b [R(x)]^2 \, dx\]

• \(R(x)\) is the radius of the disk at any point \(x\).
• The limits of integration \(a\) and \(b\) define the interval over which the solid is being revolved.

For a torus, which resembles a donut, this means revolving a region bounded by a circle around an external line, creating ring-like disks. The radius at any point \(x\) is the distance from the revolution axis to the edge of the region. By evaluating the integral for all disks across the defined interval, we can calculate the volume of the entire solid.

Solid of Revolution

A Solid of Revolution is generated by rotating a two-dimensional shape or curve around an axis. This rotation creates a three-dimensional object with symmetrical features based on the axis chosen. In the case of finding the volume of a torus, the solid is formed by revolving a circle around a line external to it, not through its center.

Typically, these problems involve:

• Determining the curve or shape to be rotated.
• Identifying the axis of revolution, which is a crucial component as it affects the structure and size of the solid.

By employing methods like the Disk Method, we can systematically compute the volume of various solids formed by such revolution. This understanding is typically applied in engineering and physics to model and calculate attributes of rotationally symmetrical objects.

Integral Calculus

Integral Calculus is a significant branch of mathematics that deals with integrals and their applications, such as calculating areas, volumes, and central points. In the context of finding the volume of a torus using the Disk Method, we apply integral calculus to sum up infinitesimally small disks filled within the solid.

Some key concepts include:

• Definite Integrals: Used to calculate the exact volume or area by integrating across given limits.
• Riemann Sums: The method by which a sum is approximated, leading to the integral's expression.

The calculation involves setting up an integral where, for each infinitesimal slice, we define its contribution to the total volume. We integrate these contributions over the specified interval, which, for the torus, is from \(-1\) to \(1\) under the integral sign. Understanding these integral principles allows us to solve real-world volume problems, particularly in understanding complex shapes and structures.