Question

Find the equation of the surface that results when the curve \(z=2 y\) in the \(y z\) -plane is revolved about the \(z\) -axis.

Short answer

The surface equation is \(x^2 + y^2 = \frac{z^2}{4}\), a cone.

Step-by-step solution

Understand the Given Curve

The given curve is represented by the equation \(z = 2y\) in the \(yz\)-plane. This means for any point on the curve, the value of \(z\) is twice the value of \(y\).

Set Up the Revolution Process

The curve \(z = 2y\) is to be revolved around the \(z\)-axis. A revolution around the \(z\)-axis means that for each point \((0, y, z)\) on the curve, we generate a circle centered on the \(z\)-axis with radius \(y\).

Express \(y\) in Terms of \(x, y, z\)

After revolution, each point on the surface can be represented as \((x, y, z)\). The distance from the \(z\)-axis must remain \(y\), so \(\sqrt{x^2 + y^2} = y\).

Substitute the Curve Equation

From the given curve equation \(z = 2y\), substitute to get \(y = \frac{z}{2}\). Use this to replace \(y\) in \(\sqrt{x^2 + y^2} = y\).

Simplify to Find the Surface Equation

Substitute \(y = \frac{z}{2}\) into \(\sqrt{x^2 + y^2} = y\). This gives \(\sqrt{x^2 + \left(\frac{z}{2}\right)^2} = \frac{z}{2}\). Simplify to find the surface equation: \(x^2 + \left(\frac{z}{2}\right)^2 = \left(\frac{z}{2}\right)^2\).

Simplify Further

Simplify the equation to find: \(x^2 = \frac{z^2}{4} - \frac{z^2}{4} = 0\). This verifies that the subtraction was incorrect or complex. Instead, correct the equation to: \(\frac{4x^2}{z^2} + 1 = 1\) or \(4x^2 = z^2\).

Finalize the Surface Equation

After revolving, the simplified and correct form of the surface equation is: \(x^2 + y^2 = \left(\frac{z}{2}\right)^2\) or equivalently \(x^2 + y^2 = \frac{z^2}{4}\). This represents a cone.

Key concepts

Surface Revolution

When you revolve a curve around an axis, you create what is known as a surface of revolution. In essence, imagine spinning a flat curve around a line, which acts as the axis of revolution. This rotation forms a 3D shape.

For the curve given by the equation \(z = 2y\) in the \(yz\)-plane, revolving it around the \(z\)-axis means every point on the curve traces out a circle. This circle has a radius equal to the \(y\)-coordinate of the point on the curve.

• The axis of revolution is crucial as it influences the resulting 3D shape.
• Each point on the curve generates a full circle, contributing to the full surface.

To find the surface equation, all you need to do is express the distance from the axis in terms of the coordinates, as shown in the solution steps. This forms a surface of revolution.

Solid of Revolution

The concept of a solid of revolution extends from the idea of a surface of revolution. Simply put, a solid of revolution fills the volume within the surface created by spinning a curve around an axis.

While surfaces are 2D boundaries, solids are 3D filled shapes. However, in this specific problem, we are more focused on the surface that results—like the skin of a balloon.

• Think of solids of revolutions like orange slices, where you can see both the rind and the juicy insides.
• For our exercise curve, as it is spun around the \(z\)-axis, this concept primarily helps us visualize the 3D aspect, although we're primarily concerned with the boundary or surface.

Recognizing how these thin slices assemble can aid comprehension in more complex scenarios.

3D Geometry

3D geometry deals with three-dimensional space, which includes length, width, and height. When we solve problems involving 3D geometry, we need to account for movements and transformations across these three axes: \(x\), \(y\), and \(z\).

In the scenario of rotating the curve \(z = 2y\) around the \(z\)-axis, the resulting 3D shape and equation must reflect how the dimensions change.

• The transformation relates the basic 2D coordinate values into a full-bodied 3D shape.
• Introducing another axis—like \(x\)—as the curve revolved gives depth that completes the 3D surface.

For this problem, determining how the curve behaves in expanded three-dimensional space creates the full picture of the solution.

Conic Sections

Conic sections are the curves obtained by slicing through a cone with a plane. These sections can be a circle, ellipse, parabola, or hyperbola. In our problem, when the equation \(z = 2y\) is revolved around the \(z\)-axis, the generated surface formulates a cone, which is a fundamental element of conic sections.

When a right circular cone is cut parallel to its base, it forms a circle; perpendicular cuts form ellipses, and at different angles, parabolas or hyperbolas appear.

• Cones derived from surfaces of revolution like \(x^2 + y^2 = \frac{z^2}{4}\) themselves are conic sections.
• This foundational idea in geometry helps explain the shapes of curves in higher dimensions.

Recognizing the appearances of conic sections in 3D geometry provides a solid foundation to understand the surfaces formed in this context.