Problem 58 - University Calculus: Early Transcendentals

Question

Use a CAS to plot the surfaces in Exercises. Identify the type of quadric surface from your graph. $$y-\sqrt{4-z^{2}}=0$$

Step-by-Step Solution

Answer

The graph shows a cylindrical surface open along the x-axis.

Step 1: Rewrite the Equation

The given equation is \( y - \sqrt{4 - z^2} = 0 \). Start by isolating \( y \):\[ y = \sqrt{4 - z^2} \]. This equation describes \( y \) as a function of \( z \).

Step 2: Identify the Surface Characteristics

The equation \( y = \sqrt{4 - z^2} \) resembles a semicircle in the \( y-z \) plane where \( x \) can take any value. More specifically, \( z^2 + y^2 = 4 \), which is a circle of radius 2 when the square root is not considered. This equation represents a surface known as a 'cylinder' along the x-axis.

Step 3: Understand the Quadric Surface

In three dimensions, the equation \( y = \sqrt{4 - z^2} \) implies that for each \( x \), there is a semi-circular arc in the \( y-z \) plane, half of a cylinder, specifically a portion of a 'circular cylinder' oriented along the x-axis.

Step 4: Use CAS to Confirm

Using a Computer Algebra System (CAS), plot the surface described by the obtained equation to visually confirm the type of surface. Note that you should see semicircular arcs forming a cylinder open to the positive \( y \)-direction along the \( x \)-axis.

Key Concepts

Circular Cylinder

A circular cylinder is one of the classic shapes found in geometry and can be better understood by visualizing basic objects like a can or tube. In a mathematical context, a circular cylinder is a surface generated by moving a circle along a straight line, known as the axis of the cylinder. Here are some key points about circular cylinders:

• The cross-section of a circular cylinder perpendicular to its axis is a circle, and it remains constant throughout the length of the cylinder.
• Circular cylinders possess symmetry and rotational uniformity around the axis.

When examining equations like the one in the exercise, where we see that the shape is defined in terms of a variable, it's important to recognize how these variables interact to form a three-dimensional object. In our case, since the equation depends only on the variables y and z, with any x-value allowed, it reflects a circular cylinder aligned along the x-axis. This type of understanding is crucial in fields ranging from engineering to computer graphics, where precise geometric modeling is essential.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is an essential tool in modern mathematics, science, and engineering fields. It provides capabilities that can help understand and visualize complex equations and geometric shapes.

• Symbolic Computation: CAS can perform algebraic manipulations, including solving equations, simplifying expressions, and factoring polynomials with exact symbolic results.
• Graphical Representation: By plotting equations, CAS helps identify and understand two-dimensional and three-dimensional shapes, like the circular cylinder in our exercise.
• Ease of Use: CAS tools are user-friendly and interactive, supporting real-time changes and immediate visual feedback to enhance learning.

Using CAS, students can input the equation directly and plot the result to see the semicircular arcs forming a cylinder. This hands-on approach is not only engaging but also deepens comprehension by relating algebraic concepts to their visual counterparts.

Semi-Circular Arc

A semi-circular arc is a simple yet significant geometric shape in many fields, forming half of a complete circle. Understanding the characteristics of a semi-circular arc enhances comprehension of more complex surfaces, like the circular cylinder.

• A semi-circular arc is defined as any portion of a circle's circumference which spans 180 degrees.
• It can be represented mathematically in the form of an equation where one of the variables is eliminated or set within certain limits, often using square root functions as seen in our case.

In the problem, the semi-circular arc appears within the y-z plane, where the equation looks like a semi-circle with a radius. When this semi-circular arc is extended along another axis, such as the x-axis in three-dimensional space, it creates the surface known as a circular cylinder. By recognizing simple forms like semi-circular arcs, students can more easily grasp the transitions to complex three-dimensional geometries.