Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

The parabola \(z=2 x^{2}\) in the \(x z\) -plane is revolved about the \(z\) -axis. Write the equation of the resulting surface in cylindrical coordinates.

Step-by-Step Solution

Answer

The equation is \( z = 2r^2 \) in cylindrical coordinates.

Step 1: Understand the Problem

We have a parabola given by the equation \( z = 2x^2 \) in the \( xz \)-plane. This parabola will be revolved around the \( z \)-axis to form a surface of revolution.

Step 2: Recall Cylindrical Coordinates

The cylindrical coordinates \( (r, \theta, z) \) relate to Cartesian coordinates \( (x, y, z) \) through the equations \( x = r\cos(\theta) \), \( y = r\sin(\theta) \), and \( z = z \).

Step 3: Setup the Revolution

When the parabola \( z = 2x^2 \) is revolved around the \( z \)-axis, each point \((x,0,z)\) traces a circle in the \( xy \)-plane at height \( z \) with radius \( r = |x| \).

Step 4: Substitute \( x \) in Terms of \( r \)

Since \( r = |x| \) and the equation of the parabola is \( z = 2x^2 \), we replace \( x^2 \) with \( (\frac{r}{cos(\theta)})^2 = r^2 \). Thus, we have \( z = 2r^2 \).

Step 5: Equation of the Surface of Revolution

In cylindrical coordinates, the resulting surface by revolving \( z = 2x^2 \) around the \( z \)-axis is given by \( z = 2r^2 \).

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates are particularly useful for problems involving symmetry around an axis, such as the axis of revolution. These coordinates consist of three values: \((r, \theta, z)\).

Here’s how each part works:

• \(r\) represents the radius from the axis (in this case, the \(z\)-axis) to the point in question. It is akin to the distance from the origin in a 2D plane, but extended into three dimensions.
• \(\theta\) is the angle around the \(z\)-axis, measured in radians. It’s similar to the angular component in polar coordinates.
• \(z\) remains the same as in Cartesian coordinates, representing the height above the \(xy\)-plane.

To translate cylindrical coordinates to Cartesian coordinates, use the following formulas:
\(x = r\cos(\theta)\), \(y = r\sin(\theta)\), and \(z = z\).
Cylindrical coordinates simplify equations involving rotations or revolution around an axis.

Parabola

A parabola is a symmetric, U-shaped curve that can open up, down, left, or right, depending on its equation and orientation. In the provided exercise, the parabola is described by the equation \( z = 2x^2 \).

In this configuration:

• The parabola opens upwards along the \(z\)-axis, as indicated by the \(2x^2\) term.
• The coefficient \(2\) impacts the width of the parabola, making it narrower than if it were simply \(x^2\).
• The vertex, or the lowest point of the parabola in the \(xz\)-plane, is located at the origin \((0, 0)\).

Parabolas are frequently encountered in physics, engineering, and design because of their property that each point is equidistant from a point (the focus) and a line (the directrix). Understanding parabolas is crucial for interpreting the equation of surfaces when they are rotated around an axis.

Z-axis Rotation

Z-axis rotation is the process of rotating a shape or curve around the \(z\)-axis. This concept is integral to forming three-dimensional surfaces from two-dimensional curves. In this context, a 2D parabola \(z = 2x^2\) is rotated to become a 3D surface.

Here's what happens during this transformation:

• Each point on the original 2D parabola sweeps out a circle around the \(z\)-axis, creating a circular path.
• The radius of each circle corresponds to the absolute value of \(x\), represented in cylindrical coordinates as \(r\).
• This cylindrical method results in a smooth surface that has rotational symmetry around the \(z\)-axis.

Such rotations are pivotal in engineering and physics when developing models of objects that have radial symmetry, such as bottles or bells. Understanding how a simple rotation can transform a planar curve helps in visualizing and creating complex designs.

Equation of Surface

Deriving the equation of a surface after revolution involves substituting variables to reflect the symmetry of cylindrical coordinates. Here, we start with a parabola \(z = 2x^2\) and desire to express it as a surface using cylindrical components.

The transformation follows these steps:

• Recognize that as \(x\) points circle around the \(z\)-axis, \(r = |x|\). Therefore, \(x^2 = r^2\) when expressed in terms of \(r\).
• Substitute \(x^2\) with \(r^2\) in the parabola's equation: \(z = 2r^2\).
• Consequently, \(z = 2r^2\) becomes the equation of the surface, representing the parabola’s revolution in cylindrical coordinates.

Understanding the equation of the surface allows one to analyze characteristics including curvature, area, and volume. It's a fundamental skill for modeling physical objects and systems in various scientific and engineering disciplines.