Problem 38 - University Calculus: Early Transcendentals

Question

Sketch the surfaces ASSORTED $$x^{2}+z^{2}=y$$

Step-by-Step Solution

Answer

The surface is a parabolic cylinder opening along the positive \(y\)-axis.

Step 1: Understand the Equation

The equation given is \(x^{2} + z^{2} = y\). This equation is a representation of a surface in a 3D space. It involves three variables: \(x\), \(y\), and \(z\).

Step 2: Identify the Type of Surface

The equation \(x^{2} + z^{2} = y\) is in the form of a paraboloid. A paraboloid opens along the axis corresponding to the variable on the right side of the equation, which in this case is the \(y\)-axis.

Step 3: Set Up Cross-Sections

To better understand how this surface looks, consider cross-sections at various values of \(y\). Fix \(y = c\), where \(c\) is a constant, and the equation becomes \(x^{2} + z^{2} = c\). For each positive value of \(c\), this describes a circle in the \(xz\)-plane with radius \(\sqrt{c}\).

Step 4: Analyze the Base Shape

When \(y = 0\), the equation reduces to \(x^{2} + z^{2} = 0\). The only solution is \(x = 0\) and \(z = 0\), implying the base of the paraboloid starts at the origin (0,0,0).

Step 5: Consider the complete shape

As \(y\) increases from 0 to positive values, the circles become larger, which shows how the paraboloid expands as it moves upward along the positive \(y\)-axis. The paraboloid is symmetrical about the \(y\)-axis due to the quadratic nature of the \(x\) and \(z\) terms.

Step 6: Sketch the Surface

Start plotting these circles on different \(y\) levels, and connect them smoothly to form a continuous surface expanding upward from the origin. Draw a vertical line representing the \(y\)-axis, and sketch the paraboloidal surface around this line.

Key Concepts

Paraboloid

A paraboloid is an important geometric surface in three-dimensional space. It is defined as a surface that can be expressed by a quadratic equation involving two variables. In this case, the equation is \( x^2 + z^2 = y \), which describes a type of paraboloid known as a circular paraboloid. Unlike its two-dimensional cousin, a parabola, a paraboloid is a curved surface. It's like flipping a bowl upside down, with the rim extending infinitely outward and the base being a single point at the origin (0,0,0).
The paraboloid in this equation opens along the \( y \)-axis, meaning it extends upwards in the direction of increasing \( y \). The form of the equation is crucial as it indicates a symmetry about the \( y \)-axis, caused by the squared terms \( x^2 \) and \( z^2 \). This symmetry gives the paraboloid a neat and consistent shape as you move along its axis of symmetry.

Cross-Sections

To comprehend the structure of a 3D surface, examining cross-sections—or slices—of the surface is incredibly helpful. Cross-sections in this context involve setting one variable constant and observing the resulting shape in the remaining dimensions.
For the equation \( x^2 + z^2 = y \), fixing the value of \( y = c \) (where \( c \) is a constant) converts the equation into \( x^2 + z^2 = c \). This is a familiar equation for a circle in the \( xz \)-plane. For different values of \( y \), these circles have different radii—specifically, \( \sqrt{c} \).
Each cross-section provides a snapshot of the surface at a specific value of \( y \). When \( y \) is positive, the cross-section is a circle centered at the origin of the \( xz \)-plane. As \( y \) increases, these circles get larger, reflecting the widening of the paraboloid as it ascends along the \( y \)-axis.

3D Graphing

Graphing a paraboloid in three dimensions can seem daunting, but breaking it down step by step makes it manageable. Start by understanding the fundamental elements of the equation \( x^2+z^2=y \). Each part represents an essential part of graphing: the symmetry around the axis, the direction in which the surface opens, and the growing size of the circles in cross-sections.
When graphed, a paraboloid appears as a smoothly curving surface stretching infinitely along one axis (

Coordinate Geometry

Coordinate geometry is the study of geometric figures using a coordinate system. It enables the translation of geometric problems into algebraic ones, providing a powerful tool in analyzing shapes like the paraboloid.
In the equation \( x^2+z^2=y \), the coordinates \( (x, y, z) \) define points on the surface. These coordinates help visualize the set of points forming the paraboloid. Here's why it's useful:

• Visualizing Shapes: Through coordinates, one can graphically depict complex shapes like a paraboloid, making it easier to understand spatial relationships.
• Identifying Symmetry: The squaring of variables \( x \) and \( z \) shows that any swap between these variables won't change the outcome, highlighting symmetry along the \( y \)-axis.
• Solving Equations: Coordinate geometry allows solving for one variable when the other two are known, aiding in constructing precise models of 3D surfaces.

With these tools, even challenging 3D structures can be tackled with clarity and precision.