Question

Give the equation of the surface of revoIution described. Revolve \(z=x^{2}\) about the \(z\) -axis.

Short answer

The surface of revolution is the paraboloid \( z = x^2 + y^2 \).

Step-by-step solution

Understand Surface of Revolution

The surface of revolution is formed when a curve is revolved around a line (the axis of revolution). Here, we are revolving the curve \( z = x^2 \) around the \( z \)-axis.

Set Up the Equation

When a curve \( z=f(x) \) is revolved around the \( z \)-axis, the surface of revolution can be described parametrically using cylindrical coordinates. The radius from the \( z \)-axis is \( \, \) \( r = x \), so \( x = r \cos \theta \), \( y = r \sin \theta \), and \( z = x^2 \), leading to the equation:

Derive the Parametric Equations

Substitute \( x = r \) into the equation for \( z \) to get \( z = r^2 \). The parametric equations describing the surface of revolution are:\(\begin{align*}x &= r \cos \theta, \y &= r \sin \theta, \z &= r^2,\end{align*}\)where \( r \geq 0 \) and \( \theta \in [0, 2\pi] \).

Eliminate the Parameters

To find the Cartesian equation, eliminate \( r \) and \( \theta \) from the parametric form. From \( x = r \cos \theta \) and \( y = r \sin \theta \), we have \( r = \sqrt{x^2 + y^2} \). Substitute \( r \) back into \( z = r^2 \):\[ z = (\sqrt{x^2 + y^2})^2 = x^2 + y^2 \]

Write the Cartesian Equation

Therefore, the equation of the surface of revolution in Cartesian coordinates is:\( z = x^2 + y^2 \)

Key concepts

Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional coordinate system that extends polar coordinates into three dimensions. Imagine wrapping a sheet around the rings of a cone. Cylindrical coordinates allow us to describe points in that space using a radius, an angle, and a height. These coordinates are represented as \(r, \, \theta, \, z\), where:

• \( r \) is the radial distance from the z-axis.
• \( \theta \) is the angle formed with a reference direction, usually the positive x-axis.
• \( z \) is the height above the xy-plane, similar to the z-coordinate in Cartesian coordinates.

By converting from Cartesian to cylindrical coordinates, it becomes easier to deal with problems involving symmetry around an axis. In our exercise, cylindrical coordinates allow us to represent the original curve \( z = x^2 \) and create a surface of revolution as it revolves around the z-axis.

Parametric Equations

Parametric equations provide a method of describing a surface or curve with one or more parameters. Rather than representing x, y, and z in terms of one another, we describe them using separate equations based on common parameters. This offers greater flexibility.
For instance, in describing the surface of revolution for the given curve \( z = x^2 \), when revolving around the z-axis, we use cylindrical coordinates that replace x and y as:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( z = r^2 \)

Here, the parameter \( r \) represents the radial distance from the z-axis, and \( \theta \) is the angle of rotation, allowing the equation to manifest as a solid three-dimensional object. Parametric equations are vital in calculus and geometry because they simplify complex surfaces and curves.

Z-Axis

The z-axis is one of the three axes in a three-dimensional space of the Cartesian coordinate system. It runs vertically and is perpendicular to both the x-axis and y-axis.
When visualizing three-dimensional shapes, the z-axis often represents the vertical component or height. In this exercise, revolving the curve \( z = x^2 \) around the z-axis means that every point on the curve will circle around this axis, creating a symmetrical form. This axis acts as a spine or central pivot, dictating the object’s symmetry in the case of a surface of revolution.
Understanding the role of the z-axis helps in visualizing and creating three-dimensional shapes based on two-dimensional curves.

Revolution

Revolution is a process in mathematics where you rotate a shape around a fixed line, called the axis of revolution. This operation generates a three-dimensional surface or solid.
In the given problem, we revolve the curve \( z = x^2 \) around the z-axis. As each point of the curve rotates 360 degrees around this axis, we form a paraboloid, a type of surface of revolution.
Key points to grasp about revolution include:

• The entire process can be illustrated well through animation, where curves elongate into surfaces.
• The resulting shape maintains symmetry around the chosen axis.
• This method is extensively used in calculus and engineering to model real-world objects like turbine blades or vases.

Revolution simplifies transforming algebraic expressions into complex three-dimensional objects, expanding their utility across various scientific and engineering fields.