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 an infinite cylinder along the \(x\)-axis with a radius of \(\sqrt{15}\) in the \(yz\)-plane.

Step 1: Understand the Equation

The given equation is \(y^2 + z^2 = 15\). There is no \(x\) variable in this equation, which implies that \(x\) can take any real value. This suggests that the graph will extend infinitely along the \(x\)-axis.

Step 2: Identify the Equation Type in 2D

The equation \(y^2 + z^2 = 15\) is similar to the equation of a circle in the \(yz\)-plane. In standard form, it can be written as \((y - 0)^2 + (z - 0)^2 = 15\), which tells us it is a circle centered at the origin \((0,0)\) in the \(yz\)-plane with a radius of \(\sqrt{15}\).

Step 3: Extend the Equation to 3D

Since the circle \(y^2 + z^2 = 15\) exists in the \(yz\)-plane for any value of \(x\), the three-dimensional graph is formed by extending this circle across all \(x\) values. This results in a cylinder extending parallel to the \(x\)-axis.

Step 4: Sketch the Graph

To sketch the graph, draw a circle in the \(yz\)-plane with radius \(\sqrt{15}\) and then extend this circle along the \(x\)-axis to form an infinite cylinder. This cylinder will have its axis aligned with the \(x\)-axis.

Key Concepts

graph of a cylinder

In the world of three-dimensional graphing, visualizing a cylinder is a fundamental concept. The equation provided, \( y^2 + z^2 = 15 \), helps us depict a cylinder. Essentially, the graph represents a cylindrical surface in 3D space. Since there is no \( x \) variable in this equation, the resulting cylinder extends infinitely along the \( x \)-axis. This axis is the lengthwise direction of the cylinder.

In practical terms, imagine a vertical circle in the \( yz \)-plane being "stretched" infinitely along the \( x \)-axis. This creates a shape that can be visualized as a pipe or a tunnel. The absence of \( x \) in the equation means any \( x \) value is permissible, confirming the cylinder's infinite length.

transforming equations

Equation transformation is crucial in identifying the types of shapes described by mathematical expressions. By recognizing how to transform equations, we can understand the geometry they imply. Here, the equation \( y^2 + z^2 = 15 \) is rearranged to \((y - 0)^2 + (z - 0)^2 = 15\). This transformation helps us notice it's a circle equation.

• \( y^2 + z^2 = r^2 \) — This standardizes the circle's equation, where \( r \) is the radius.
• Here, \( r = \sqrt{15} \), hinting at the circle's size in the \( yz \)-plane.

Recognizing such transformations aids in graphing complex shapes in 3D, making concepts more intuitive and understandable.

plane coordinates

Coordinate planes are pivotal in graphing and understanding shapes in three dimensions. For the given problem, we consider the \( yz \)-plane specifically.

• This plane is where we visualize the circle first, before extending it into three-dimensional space.
• The \( yz \)-plane consists of the vertical and depth dimensions, ignoring height or horizontal lengths like \( x \).

To better understand, it's like looking at a two-dimensional slice of a three-dimensional object. The origin point \((0, 0)\) is where this circle centers in the \( yz \)-plane, vital for graphing our cylinder correctly.

three-dimensional circle

When moving from a two-dimensional circle to three-dimensional space, the concept can be confusing. The equation \( y^2 + z^2 = 15 \) represents a circle in the \( yz \)-plane

• Center: Origin \((0, 0)\) in \( yz \)-plane.
• Radius: \( \sqrt{15} \).

By extending this circle with respect to the \( x \)-axis, the discussion transitions to a three-dimensional context, facilitating the visualization of an infinite cylinder. In 3D graphing, writing down such details allows learners to transition smoothly from basic circles to complex surfaces seamlessly.