91Ó°ÊÓ

Parametric equations are a pair (or set) of equations that express a set of quantities as explicit functions of a number of independent variables, known as parameters. The parametric equation for curves in two dimensions can be written as \(x(t) = a(t)\) and \(y(t) = b(t)\). For example, a circle in the plane can be described using parametric equations with \(x(t) = r\cos(t)\) and \(y(t) = r\sin(t)\).
When dealing with surfaces of revolution, parametric equations become handy because they can express complex three-dimensional shapes easily. For instance, if you revolve a curve \(y = f(x)\) around an axis, the resulting surface can be represented in parametric form. Here’s how it works:

• You start by identifying the curve's equation.
• Then, introduce parameters to describe points on the surface.
• The parametric equation for the surface of revolution incorporates the original two-dimensional curve and adds a new dimension of rotation.

In our problem, the original curve \(x^2 = 4y\) is parametric in terms of \(x\) and \(y\), and becomes parametric in three dimensions when you introduce \(z\) to describe the rotation about the x-axis.

A parabola is a type of curve on a plane, defined by a quadratic equation of the form \(y = ax^2 + bx + c\). Commonly, it appears in equations like \(x^2 = 4y\) or \(y = x^2\). The equation \(x^2 = 4y\) represents a parabola that opens upwards with its vertex at the origin (0,0).
Here are some features:

• The vertex: The turning point of the parabola.
• The axis of symmetry: A vertical line that passes through the vertex.
• The focus: A point inside the parabola where rays parallel to the axis of symmetry reflect off the surface.

When we revolve a parabola around an axis, we get a three-dimensional object. In our case, the parabola \(x^2 = 4y\) when revolved around the \(x\)-axis forms a surface called a paraboloid. This transformation involves rotating every point on the parabola 360 degrees around the \(x\)-axis, producing a symmetrical shape in three dimensions.

Three-dimensional geometry extends our understanding of shapes to three-dimensional space with coordinates \(x, y, z\). In this space, curves and surfaces can be more complex and exciting.
When a curve is revolved around an axis, it creates a surface of revolution. To find such a surface mathematically:

• Identify the curve’s equation in two dimensions.
• Apply the distance formula to the rotated form to introduce a third dimension.
• The resulting equation will describe the surface of revolution in 3D space.

In the given exercise, we started with \(x^2 = 4y\) and revolved it around the \(x\)-axis. To incorporate the third dimension, we used \(z\) as the revolution axis and converted the equation into parametric form, \((x, y, z)\). Finally, combining all this gives us the surface equation \(y^2 + z^2 = \frac{x^2}{4}\). This describes a paraboloid in three-dimensional space.
Visualizing such shapes can be simplified with sketches. For instance, draw the original two-dimensional curve and illustrate the rotation to help you understand how it transforms into a 3D surface.