Problem 19 - APEX Calculus

Question

In Exercises 19-22, give the equation of the surface of revoIution described. Revolve \(z=\frac{1}{1+y^{2}}\) about the \(y\) -axis.

Step-by-Step Solution

Answer

The equation is \(x^2 + z^2 = \left(\frac{1}{1+y^2}\right)^2\)."

Step 1: Understand the Problem

We have a function of the form \(z = \frac{1}{1+y^2}\) and the task is to create a surface of revolution by revolving this curve around the \(y\)-axis.

Step 2: Use The Formula for Surface of Revolution

The general formula for the surface of revolution obtained by rotating a curve \(z=f(y)\) about the \(y\)-axis is \(x^2 + z^2 = f(y)^2\), where \(x\) is the horizontal distance from the axis of rotation.

Step 3: Substitute the Given Function

Substitute \(z = \frac{1}{1+y^2}\) into the formula for the surface of revolution: \[x^2 + z^2 = \left(\frac{1}{1+y^2}\right)^2\].

Step 4: Express the Equation

Since \(z\) in the surface equation has already been defined as \(z = \frac{1}{1+y^2}\), the equation becomes \[x^2 = \left(\frac{1}{1+y^2}\right)^2 - z^2\].

Step 5: Simplify the Equation

This equation defines the surface in terms of \(x\), \(z\), and \(y\), expressing the shape of the surface. Simplification is not required beyond the context of specifying the resulting surface equation.

Key Concepts

Equation of a Surface

When dealing with surfaces of revolution, we're often tasked with finding the mathematical representation—a precise equation—that defines the surface. In simple terms, an equation of a surface helps to describe every point on that surface in a mathematical way. For instance, given a specific curve or function, like \( z = \frac{1}{1+y^2} \), we want to revolve this curve about an axis, here specifically the \( y \)-axis, to generate a full 3D surface.

• The general goal is to form a new equation that encapsulates the complete surface formed by such a rotation.
• The equation is typically expressed in terms of variables that represent spatial dimensions, such as \( x, y, \) and \( z \).
• This equation is crucial for visualizing, analyzing, and further calculating properties of the surface, such as its area or volume.

By recreating the equation step-by-step, we develop a deeper understanding of how a simple curve can transform into a complex, three-dimensional object.

Rotation About an Axis

Rotation about an axis is like spinning an object around a fixed line. In our exercise, the curve described by \( z = \frac{1}{1+y^2} \) is revolved around the \( y \)-axis.

• Think about it as taking a flat shape and spinning it around a vertical line. This spinning creates a volume of revolution.
• The axis of rotation acts as the center of this spinning motion.

Consider how a point on the curve, located at a specific \( y \)-value, moves in a circular path around the axis. This motion defines a ring or circular disc, which accumulates together to form the entire surface of revolution. By integrating these small rotations, we can visualize and mathematically define new surfaces.

Curve Representation

Understanding the curve representation is essential for creating the surface of revolution. In our example, the curve \( z = \frac{1}{1+y^2} \) is a mathematical expression that describes a specific shape in two dimensions before it is revolved.

• The curve starts as a simple function, a visualized line or curve in a 2D plane.
• This function expresses the dependency between \( z \) and \( y \), indicating how \( z \) changes as \( y \) varies.

By clearly understanding this original curve as a slice of the upcoming 3D shape, it allows us to predict how it will unfold once rotated. Once we understand this slice, it becomes straightforward to extend it around the axis, producing a complete surface, leading to the final equation structure. Recognizing this start-to-end process is crucial for mastering the concept of surfaces of revolution.