Problem 41 - Calculus with Concepts in Calculus

Question

Sketch the quadric surface. \(y^{2}+z^{2}=9\)

Step-by-Step Solution

Answer

It's a circular cylinder along the x-axis with radius 3 in the yz-plane.

Step 1: Recognize the Equation Type

The equation given is in the form of \[ y^2 + z^2 = 9 \]This is the standard form of a cylinder, specifically a circular cylinder, because it only involves the variables \(y\) and \(z\). The term missing is the \(x\) variable, implying the circular cross-section's invariance along the \(x\)-axis.

Step 2: Identify the Shape of Cross-sections

Examine the equation \[ y^2 + z^2 = 9 \]This form represents a circle with radius 3 in the \(yz\)-plane (axes \(y\) and \(z\)). Every cross-section parallel to the \(yz\)-plane is a circle of radius 3.

Step 3: Determine the Direction of the Cylinder

Since the equation does not involve the variable \(x\), the surface described is a cylinder that extends infinitely along the \(x\)-axis. Therefore, for every value of \(x\), the cross-section is the same circle.

Step 4: Sketch the Surface

In the sketch, draw the cross-section circle with radius 3 in the \(yz\)-plane centered at the origin \((y, z) = (0, 0)\). Then replicate these circles along the \(x\)-axis in both directions, showing that the cylinder is extended along the \(x\)-axis. The result is an infinitely long cylinder with circular cross-sections.

Key Concepts

Circular Cylinder

A circular cylinder is a three-dimensional surface formed by moving a circle in a straight line through space. In the context of the equation \( y^2 + z^2 = 9 \), this movement occurs along the \(x\)-axis, since the equation lacks an \(x\) term. The essence of a circular cylinder can be described as:

• The circle remains constant in shape and size; this one has a radius of 3.
• The circles stack infinitely along the given direction (here, the \(x\)-axis).

This concept is useful because it lets us visualize complex 3D objects through a simple repeated shape. Unlike other surfaces that might twist or curve more dramatically, circular cylinders maintain the same cross-section all the way through.

Cross-sections

Cross-sections are slices or views of a three-dimensional object taken parallel to a plane. For the equation \( y^2 + z^2 = 9 \), each cross-section in the \(yz\)-plane is crucial to understanding the surface. What you'll see is a circle:

• Each circle has a radius of 3.
• The center of these circles is at the origin, \((0,0)\), on the \(yz\)-plane.

This means all visual slices of this cylinder, taken parallel to the \(yz\)-plane, appear identical. Each circle implies a perfectly round cut through the cylinder, aiding in picturing how the cylinder extends through space along the \(x\)-axis. Recognizing these cross-sections allows us to grasp the full 3D shape by understanding just one consistent two-dimensional form.

Equation Type

The equation \( y^2 + z^2 = 9 \) belongs to a specific group of equations that describe quadric surfaces. Quadric surfaces are second-degree equations in three variables (\(x\), \(y\), and \(z\)). Despite their variety, the absence of one variable simplifies interpretation:

• The absence of the \(x\) term indicates that the surface extends along the \(x\)-axis.
• The remaining equation, \( y^2 + z^2 = 9 \), depicts a set of circles.

This particular equation is known as a circular cylinder. The consistent structure of the equation, involving only the squares of two variables, reveals two key aspects: the shape of cross-sections (circles) and the direction of extension. Recognizing the equation type simplifies the process of sketching and understanding its geometry.