Problem 32 - Calculus Early Transcendentals

Question

When the circle \(x^{2}+(y-a)^{2}=r^{2}\) on the interval \([-r, r]\) is revolved about the \(x\) -axis, the result is the surface of a torus, where \(0

Step-by-Step Solution

Answer

Question: Given the equation of a circle as \(x^{2}+(y-a)^{2}=r^{2}\), where \(a\) is the distance from the center of the circle to the x-axis and \(r\) is the radius of the circle, show that the surface area of the torus formed by revolving the circle around the x-axis is \(S = 4 \pi^{2} a r\).

Step 1: Express y in terms of x

We need to solve the circle equation for \(y\), so that we can use it later to find the radius of revolution. That means we'll isolate \(y\) on one side of the equation: \(x^{2}+(y-a)^{2}=r^{2}\) Add \((-x^2)\) to both sides \((y-a)^{2}=r^{2}-x^{2}\) Take the square root of both sides: \(y-a = \pm\sqrt{r^{2}-x^{2}}\) Now, add \(a\) to both sides: \(y = a \pm\sqrt{r^{2}-x^{2}}\)

Step 2: Find the radius of revolution

The radius of revolution, \(R\), is the vertical distance between the x-axis and the center of the circle, which is \(a\). To find this distance, we analyze the circle equation and observe that the center of the circle is at \((0, a)\). Thus, the radius of revolution is simply \(R = a\).

Step 3: Calculate the surface area of the torus

Let's plug in the value of \(R\) we found in the last step into the formula for the surface area of a torus, and simplify: \(S = 2 \pi R \cdot 2 \pi r = 2 \pi a \cdot 2 \pi r = 4 \pi^{2} a r\) So, the surface area of the torus is indeed \(S = 4 \pi^{2} a r\).

Key Concepts

Revolution of Curves

The idea behind the revolution of curves is quite fascinating. Imagine taking a shape, like a circle, and spinning it around an axis. This spinning action creates a three-dimensional object. In our case, we take a circle and revolve it around the x-axis to create a torus.

Visualizing this can be like watching a doughnut forming, with the path of the circle, when spun, creating that familiar hole in the middle. It's all about creating volume and surface area in three dimensions from a two-dimensional curve.

This concept is essential in calculus and engineering because it allows us to calculate properties like volume and surface area of complex shapes. In particular, when we revolve curves, we often apply integration techniques to determine these properties and understand how the shape behaves in space.

Circle Equation

The equation of a circle is a fundamental concept in geometry. For a circle centered at a point \(0, a\) with a radius of \(r\), the equation is given by \(x^{2}+(y-a)^{2}=r^{2}\).

This equation expresses the set of all points (x, y) that are equidistant from the center (0, a). To solve for \(y\), you rearrange terms to isolate it: \(y = a \pm\sqrt{r^{2}-x^{2}}\).

Understanding the circle equation is vital because it lets us analyze and manipulate the circle's position, size, and orientation. In problems involving the revolution of curves, like finding a torus's surface area, the circle equation provides the foundation for further calculations.

Radius of Revolution

The radius of revolution is key to understanding how 2D shapes transform into 3D objects. For the torus, the radius of revolution is the distance from the axis of revolution to the center of the original rotating shape, in this example, the circle.

In our scenario, the center of the circle is located at \(0, a\), which means that the radius of revolution is simply \(a\). This radius acts like the handle around which the circle rotates.

• The radius \(a\) determines the distance of the center of revolution from the axis.
• This distance is crucial because it influences the size and shape of the resulting torus.

By knowing the radius of revolution, we can accurately compute the surface area using the formula \(S = 4 \pi^2 a r\), capturing how the circle's spin forms the torus.