Problem 53 - Essential Calculus

Question

Find the surface area of the torus generated by revolving the circle given by \(r=2\) about the line \(r=5 \sec \theta\)

Step-by-Step Solution

Answer

The surface area of the torus is \(20 \pi^2\).

Step 1: Identify the radius of the circle and distance from the line

Given the radius \(r=2\) and the distance from the line \(R=5\) obtained from the equation \(r = 5 \sec \theta\).

Step 2: Insert the values into the formula for torus surface area

Use the formula for the surface area of a torus which is \(2 \pi^2 R r\), and replace \(R\) with 5 and \(r\) with 2. This will give: \(2 \pi^2 * 5 * 2\).

Step 3: Compute the surface area

Carry out the multiplication in the equation from Step 2, resulting in: \(2 * \pi^2 * 5 * 2 = 20 \pi^2\).

Key Concepts

Solid of Revolution

Understanding the concept of a solid of revolution is essential when venturing into three-dimensional calculus problems. Simply put, a solid of revolution is a three-dimensional object created by rotating a two-dimensional shape around an axis.

The torus is a classic example of a solid of revolution. Imagine taking a circle, like a rubber ring, and spinning it around a line outside the circle. This action sweeps out a donut-shaped object known as a torus. To create this shape mathematically, we use parameters such as the radius of the circle and the distance from the axis of revolution.

Creating a solid of revolution isn't just a theoretical exercise; it has practical applications in engineering and design, where such shapes are common. The beauty of calculus is that it provides the tools to calculate properties like volume and surface area of these complex shapes.

Surface Area Formula

When dealing with the surface of three-dimensional objects created via rotation like a torus, the surface area formula plays a pivotal role. For the torus, the surface area formula is typically expressed as \(2 \pi^2 R r\), where \(R\) is the distance from the center of the tube to the center of the torus, and \(r\) is the radius of the tube.

Breaking Down the Torus Surface Area Formula

Let's dissect the torus surface area formula to see why it works. The \(2\pi r\) part represents the circumference of the circle being rotated, and the \(2\pi R\) represents the path that the circle's center takes during the rotation. Multiplying these two circumferences together gives us the total surface area.

By using this formula, we can quickly find the surface area of complex shapes, enhancing our understanding of geometry and furthering the practical application of calculus in various fields.

Calculus Problems

Solving calculus problems often involves using formulas and understanding the concepts behind them. The issue with simply memorizing formulas is that it may not help when faced with a new or complex problem. Instead, knowing the principles and reasoning behind the formulas are what enable problem solvers to approach calculus challenges effectively.

Applying Concepts to New Situations

In our exercise, we executed steps that show the practical use of calculus in finding the surface area of a torus. This starts with identifying all given parameters and then applying the appropriate surface area formula, demonstrating a real-world application of calculus problem-solving.

With practice, the problem-solving skills honed in calculus can be applied to a variety of situations, from physics to economics. It's all about identifying patterns, understanding relationships, and using mathematical tools to find solutions. The beauty of solving calculus problems lies in the ability to break down complex tasks into simpler steps, thereby making the problem less daunting.