Problem 39 - University Calculus: Early Transcendentals

Question

Sketch the surfaces ASSORTED $$x^{2}+z^{2}=1$$

Step-by-Step Solution

Answer

The surface is a vertical cylinder with a circular cross-section in the xz-plane.

Step 1: Identify the Equation Type

The given equation is \(x^2 + z^2 = 1\). This is an equation in two variables (x and z), and it represents a circle in the xz-plane, centered at the origin with a radius of 1.

Step 2: Extend the Equation to Three Dimensions

In three-dimensional space, this circle describes a cylindrical surface, as there is no constraint on the variable \(y\). This means that the circle extends infinitely in the y-direction.

Step 3: Sketch the Base Circle

To sketch the cylindrical surface, start by drawing the base circle in the xz-plane. This circle has a center at the origin \((0, 0)\) and is defined by the equation \(x^2 + z^2 = 1\), forming a perfect circle.

Step 4: Draw the Cylinder

Extend the circle vertically along the y-axis. Since there is no limitation on the height of the cylinder, it goes from \(-\infty\) to \(\infty\) along the y-direction, creating an infinite cylindrical surface.

Step 5: Finalize the Sketch

Ensure the cylindrical surface shows the circular base and extends vertically. It may be helpful to shade or use dotted lines to indicate the extension along the y-axis, emphasizing the three-dimensional nature of the surface.

Key Concepts

Equation in Two Variables

An equation in two variables is a mathematical expression where two different variables are related to one another. In the context of this exercise, we have the equation \(x^2 + z^2 = 1\). This type of equation typically describes geometric shapes when graphed. Here, we observe that the expression \(x^2 + z^2 = 1\) forms a circle, as it involves the sum of the squares of two variables.In the

• xz-plane,
• this circle is centered at the origin (point (0, 0)).
• The radius is 1 unit, since the equation equals 1.

This equation does not include the variable \(y\), hence there is no restriction in the third dimension, which is vital when we extend it to three-dimensional applications.

Three-Dimensional Sketching

When extending an equation from two dimensions to three, the third variable becomes crucial. In the given scenario, the equation \(x^2 + z^2 = 1\) defines a circle in the xz-plane. For three-dimensional sketching, consider how the circle interacts with the unrestricted variable \(y\).Since there is no \(y\) term in the equation, the shape is free to extend along the \(y\)-axis. This means the circle forms the base of an infinite cylindrical surface.To sketch this:

• Draw the circle described by \(x^2 + z^2 = 1\) in the xz-plane.
• Visualize this circle extending upward and downward in the y-direction, creating a tube-like structure, or cylinder.
• Show the direction of extension using vertical dashed or shaded lines for clarity.

This draws attention to the infinite nature of the cylinder along the \(y\)-axis.

Coordinate Plane

Understanding coordinate planes is fundamental in graphing and interpreting geometric figures. A coordinate plane consists of two axes that define a two-dimensional surface. In this exercise, the xz-plane is the relevant coordinate plane.

1. The xz-plane forms the base where the circle from our equation is plotted.
2. Here, each point on the circle satisfies \(x^2 + z^2 = 1\).
3. The center of the circle is at the origin (0,0) in the xz-plane.

By recognizing the xz-plane as the starting point for our sketch, it makes handling three-dimensional surfaces more intuitive. Visualize the xz-plane as a flat surface where the initial geometry is formed before extruding it along the y-axis to create the cylinder. This concept underscores the transition from a two-dimensional shape to a three-dimensional form seamlessly.