Problem 6 - Calculus: Early Transcendentals

Question

Suppose the curve \(z=x\) in the \(x z\) -plane is rotated around the z-axis. Find an equation for the resulting surface in cylindrical coordinates.

Step-by-Step Solution

Answer

The equation for the surface in cylindrical coordinates is \(z = r\cos(\theta)\).

Step 1: Understanding the Exercise

We need to find the equation of a surface generated by rotating the curve \(z = x\) around the z-axis. This means for every point on the curve, we will trace a path around the z-axis to create a 3D object.

Step 2: Convert Cartesian to Cylindrical Coordinates

In cylindrical coordinates, a point is represented as \((r, \theta, z)\) where \(r\) is the radial distance from the z-axis, \(\theta\) is the angular coordinate, and \(z\) is height. A point \((x, y, z)\) in Cartesian coordinates can be converted to cylindrical coordinates by using \(r = \sqrt{x^2 + y^2}\), \(\theta = \text{atan2}(y, x)\), and \(z = z\). However, since we are rotating around the z-axis, the angle \(\theta\) can vary from 0 to \(2\pi\).

Step 3: Equation for Curve in Cylindrical Coordinates

The original line \(z = x\) expressed in cylindrical coordinates will give us \(z = r \cos(\theta)\) because when the curve is rotated around the z-axis, the x-coordinate becomes \(r\cos(\theta)\). However, since the curve lies entirely in the xz-plane, the y-coordinate is 0, and after rotation, \(r = \sqrt{x^2 + 0^2} = |x|\).

Step 4: Establish the Surface Equation

The curve \(z = x\) implies in cylindrical coordinates \(z = r\cos(\theta)\) when rotated around the z-axis. Since this applies for any angle \(\theta\), the equation for the surface in cylindrical coordinates becomes \(z = r\cos(\theta)\).

Key Concepts

Surface Equation

The task in the exercise is to convert the curve represented by the equation \(z = x\) into a 3D surface through rotation around the z-axis. When a curve is rotated about an axis, every point on the curve creates a circular path, forming a surface.
This surface is called a surface of revolution, and its equation can be derived from the initial curve.In our specific problem, the curve starts as a simple line \(z = x\) in the 2D xz-plane. As we rotate this line around the z-axis, each point transforms into a circle parallel to the xy-plane at a certain height \(z\).
The collection of these circles as \( \theta \) moves from 0 to \(2\pi\) will trace out the entire surface.The finished surface equation in cylindrical coordinates becomes \(z = r\cos(\theta)\). This equation describes a full, symmetrical surface around the z-axis, simplifying the complex 3D geometry into a neat formula.

Coordinate Conversion

Converting between coordinate systems is a fundamental skill in math and physics. It allows us to view problems from different perspectives, often simplifying complex situations.

Cylindrical coordinates are particularly useful for problems with rotational symmetry, like the one in the exercise, because they naturally incorporate angles.A point in 3D space can be expressed in two common systems:- **Cartesian coordinates**: \((x, y, z)\)- **Cylindrical coordinates**: \((r, \theta, z)\), where: - **\(r\)** is the distance from the point to the z-axis - **\(\theta\)** is the angle with the positive x-axis - **\(z\)** is the heightThe conversion formulas are:- \(r = \sqrt{x^2 + y^2}\)- \(\theta = \text{atan2}(y, x)\)- \(z = z\)In this exercise, we use these conversions to express the line equation \(z = x\) in cylindrical terms.
The x-component becomes \(r \cos(\theta)\), and since \(y = 0\) along the xz-plane, we have \(r = |x|\).
Therefore, the cylindrical variation maintains \(z = r\cos(\theta)\), seamlessly transforming the 2D line into a full 3D surface.

3D Rotation

3D rotation is a process of turning the figure around an axis—a concept widely useful in fields like animation, physics, and engineering.
In this exercise, rotating a line around the z-axis generates a surface of revolution.When we rotate around the z-axis:- **Plane figures**: translate into surfaces- **Lines**: transform into cylindrical shapes (around the z-axis)In cylindrical coordinates, \(\theta\) handles the rotation by defining the planar angle. As \(\theta\) varies from 0 to \(2\pi\), we observe a full rotation spanning 360 degrees.
Thus, any element of the original shape that lies a certain distance from the z-axis is duplicated along a circular path.This rotation transforms the original line \(z = x\) into a surface by extending each point along a circular trajectory in the xy-plane.
The final geometric form is more intuitively managed through cylindrical coordinates, where \(r\) and \(\theta\) provide a clear view of the radial displacement and angular rotation. This makes understanding the resulting 3D shape simpler and more mathematical.