Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name and sketch the graph of each of the following equations in three-space. $$ y^{2}+z^{2}=15 $$

Step-by-Step Solution

Answer

The graph is a circular cylinder centered along the x-axis with radius \(\sqrt{15}\).

Step 1: Understand the Equation

The equation given is \(y^2 + z^2 = 15\). This looks like an equation of a cylinder. In three-space, when an equation is in the form \(y^2 + z^2 = r^2\), where \(r^2\) is a constant, it represents a cylindrical surface centered around the x-axis with radius \(r\).

Step 2: Identify the Type of Graph

Based on the equation \(y^2 + z^2 = 15\), we identify this as a circular cylinder because it is centered along the x-axis where \(y\) and \(z\) are present together in a circular form and \(x\) does not appear in the equation.

Step 3: Determine the Radius

To find the radius of the cylinder, compare it to \(y^2 + z^2 = r^2\). Here, \(r^2 = 15\), thus the radius \(r = \sqrt{15}\).

Step 4: Sketch the Graph

To sketch the graph of the cylinder: 1. Recognize that the cylinder is infinite along the x-axis.2. For any fixed value of x, the graph in the yz-plane should be a circle with radius \(\sqrt{15}\).3. Draw several circles along the x-axis with radius \(\sqrt{15}\), illustrating the cylindrical shape expanding along the x-axis.

Step 5: Name the Graph

The final graph is a circular cylinder. It extends infinitely in both directions along the x-axis, with circular cross-sections in the yz-plane, each having a radius of \(\sqrt{15}\).

Key Concepts

Equation of a Cylinder

Understanding the equation of a cylinder is key to grasping three-dimensional shapes. In three-dimensional space, a cylinder's equation typically takes the form \(y^2 + z^2 = r^2\). Here, \(r^2\) is a constant that depicts the squared radius of the cylinder. Unlike a circle in two dimensions, this equation does not contain the variable \(x\). Instead, it implies that the cylinder stretches infinitely along the x-axis.

The equation \(y^2 + z^2 = 15\) is an example of a cylinder's equation. This representation clearly signifies a circular cross-section, with its axis along the x-direction. A cylinder can be understood as a series of stacked circles, all sharing an axis and having the same radius, thus forming a "tube-like" shape in space.

Three-Dimensional Graphing

Graphing in three dimensions can initially seem daunting, but it's a powerful tool for visualizing complex equations. When considering the cylinder like \(y^2 + z^2 = 15\), we think of it in the context of three-space.

• Visualize the x-axis as the direction along which the cylinder extends infinitely. This implies that no matter the x value, the circular cross-section's characteristics remain the same.
• In the yz-plane, where x is held constant at any value, the graph appears as a circle with the determined radius.

To graph, imagine drawing several such circles at various x values; connecting these circles creates the idea of the cylinder stretching throughout the space.

Circular Cross-Section

The concept of a circular cross-section is crucial for understanding cylinders. A cross-section is essentially a two-dimensional "slice" of a three-dimensional object. For the equation \(y^2 + z^2 = 15\), every cross-section parallel to the yz-plane is a circle.

Here’s how it works:

• Each circle has the same radius, \(\sqrt{15}\).
• Because the equation lacks an x-component, each cross-section appears identical regardless of the x value chosen.

These identical circles combine to form the surface of the cylinder. A practical way to imagine this is by considering the construction of cylinders from stacked, identical disks or rings.

Radius Calculation

Determining the radius of a cylinder when given an equation is a straightforward process. In the context of the equation \(y^2 + z^2 = r^2\), calculating the radius requires only taking the square root of the constant on the right-hand side.

For the equation \(y^2 + z^2 = 15\):

• The right side is 15, which means \(r^2 = 15\).
• Thus, the radius \(r\) is \(\sqrt{15}\).

This radius is crucial for constructing the circular cross-sections discussed earlier. It determines the size of each circle that forms the cylinder in the yz-plane. Simple mathematical manipulation, like computing a square root, often unveils the fascinating geometrical attributes of three-dimensional objects.