Problem 30 - Calculus with Concepts in Calculus

Question

Sketch the graph of the equation. \(x=\sqrt{4-y^{2}-z^{2}}\)

Step-by-Step Solution

Answer

The graph is the right hemisphere (\(x \geq 0\)) of a sphere with radius 2 centered at the origin.

Step 1: Understand the Equation

The equation given is \(x = \sqrt{4 - y^2 - z^2}\). This is a three-dimensional equation involving the variables \(x\), \(y\), and \(z\). It represents a surface in 3D space.

Step 2: Recognize the Form

The equation \(x^2 = 4 - y^2 - z^2\) can be rearranged to the form \(x^2 + y^2 + z^2 = 4\). This is the equation of a sphere centered at the origin with radius 2.

Step 3: Consider the Domain of x

Since \(x = \sqrt{4 - y^2 - z^2}\), \(x\) can only be non-negative. This means even though the full sphere is represented by \(x^2 + y^2 + z^2 = 4\), our original equation only represents the half-sphere where \(x \geq 0\).

Step 4: Sketching the Graph

The graph of the equation will be the right (positive \(x\)-axis) hemisphere of the sphere centered at the origin with radius 2. It includes all points on the sphere’s surface where \(x\) is greater than or equal to 0.

Key Concepts

Understanding 3D Coordinates

Three-dimensional (3D) coordinates are a way of locating points in space using three numbers:

• The first number, usually referred to as the x-coordinate
• The second number, the y-coordinate
• The third number, the z-coordinate

Each of these corresponds respectively to the point’s position along the horizontal, vertical, and depth axes.
3D coordinates are vital for visualizing and working with shapes and figures in space. They extend the basic 2D coordinate system by adding depth, which allows us to plot points and draw figures like spheres, cubes, and cylinders more precisely. For example, a point with 3D coordinates (2, 3, 5) is located 2 units along the x-axis, 3 units along the y-axis, and 5 units along the z-axis. This point exists in a 3D space where visualizing trajectories, surfaces, or volumes is possible.

Deriving the Sphere Equation

The sphere equation plays a crucial role in understanding complex spatial figures. While you might know circles from 2D space equations, spheres are the 3D counterparts. The general form of a sphere’s equation is \[x^2 + y^2 + z^2 = r^2\],where \\(r\) is the radius of the sphere. The equation describes all points \\((x, y, z)\) that are at a distance \\(r\) from the center of the sphere, typically located at the origin (0, 0, 0).
In the context of our exercise, the rearranged equation \\(x^2 + y^2 + z^2 = 4\)\,indicates a sphere with a radius of 2, centered at the origin.
The sphere equation serves as a foundational concept in understanding 3D shapes, representing an infinite set of points in 3D space equidistant from a center point. This equation helps in visualizing solid shapes like the ones seen in graphical and real-world applications, such as modeling planets, molecules, or any other spherical objects.

Domain Restrictions on 3D Graphing

When dealing with 3D graphs, domain restrictions often come into play. These restrictions limit the range of values certain variables can take, based on realistic or problem-specific requirements. In our exercise, the equation \\(x = \sqrt{4 - y^2 - z^2}\)limits \\(x\) to non-negative values to ensure mathematical validity.
The square root function \\(\sqrt{}\)only outputs non-negative results, thus forcing \\(x\) in our equation to be zero or positive.
This domain restriction means that instead of a full sphere, we visualize only the section where \\(x\) is more than or equal to 0—essentially, the right hemisphere of the sphere. Domain restrictions like this one guide how we interpret and graph mathematical equations and are essential for deriving accurate visuals and solutions in 3D modeling.