Problem 57 - Thomas Calculus

Question

Sketch the surfaces in Exercises \(13-76\) $$ x^{2}+z^{2}=1 $$

Step-by-Step Solution

Answer

The surface is an infinite circular cylinder along the y-axis with a radius of 1.

Step 1: Recognize the Equation Type

The equation \(x^2 + z^2 = 1\) is a standard form of a cylinder equation in the 3D space where \(x\) and \(z\) are related, while \(y\) is not involved in the equation.

Step 2: Identify the Known Shape

The equation \(x^2 + z^2 = 1\) describes a circle in the 2D \(xz\)-plane with a radius of 1. This circle is centered at the origin (0, 0) in the \(xz\)-plane.

Step 3: Extend the Shape along the y-axis

Since \(y\) does not appear in the equation, it indicates that the circle in the \(xz\)-plane extends infinitely along the \(y\)-axis, forming a circular cylinder.

Step 4: Sketch the Cylinder

To sketch this surface, draw a circle of radius 1 in the \(xz\)-plane and then imagine extending this circle along the \(y\)-axis in both directions to form an infinite cylinder. This cylinder's shape does not change regardless of the \(y\)-coordinate.

Step 5: Conclusion and Surface Characteristics

The sketched surface is a right circular cylinder with its axis along the \(y\)-axis. Its radius is 1, and it extends infinitely in the positive and negative \(y\)-direction, maintaining a consistent circular cross-section.

Key Concepts

Cylindrical Surfaces

Cylindrical surfaces are integral parts of 3D geometry. They are created when a curve in a plane is extended along an axis perpendicular to that plane. In essence, if you take a 2D shape, like a circle, and extend it straight along an axis that does not lie in the plane of the shape, you end up forming a cylinder. To visualize this, think of wrapping a rectangle around to create a circular shape, giving you a tube-like structure. The most common cylindrical surface is the **circular cylinder**. However, cylinders can take on different cross-sectional shapes.

• **Elliptic cylinders** have ellipses as cross-sections.
• **Parabolic cylinders** have parabolas as cross-sections.

In these cylinders, the axis along which the shape is extended is called the **axis of the cylinder**. For the cylinder formed by the equation given in the problem, this axis is the line that runs parallel to the y-axis.

Equation of a Cylinder

The equation of a cylinder typically arises from the need to describe its surface in three-dimensional space. A basic form of this equation for a circular cylinder whose axis is aligned along a coordinate axis is relatively straightforward. For instance, when the axis is along the y-axis:The generic equation for this type of cylinder can be expressed as:\[x^2 + z^2 = r^2\]Where

• \(x\) and \(z\) are the variables that interact to form circles at different heights along the y-axis.
• \(r\) represents the radius of these circles.

In the example provided in the exercise, the formula \(x^2 + z^2 = 1\) indicates that we are dealing with a circular cylinder with a radius of 1.

• The absence of the \(y\)-variable shows that the circle in the \(xz\)-plane repeats for all values of \(y\).

This leads to an infinitely extending surface parallel to the y-axis. By understanding the benefits of this simplified form, one can easily interpret the essential characteristic of aligned cylinders in space.

Surface Sketching

Surface sketching in 3D geometry can seem daunting initially, but by following a few systematic steps, the complexity can be mitigated. When sketching a cylindrical surface, start by visualizing its base shape in two dimensions. In this case, it is a circle in the \(xz\)-plane.

• **Step 1**: Draw the circle at the origin, which is where both the \(x\) and \(z\) coordinates are zero.
• **Step 2**: Extend this circle along the axis not involved in its equation, which is the \(y\)-axis in our example.

Imagine stacking one circle on top of another in the \(y\)-direction, thus creating the illusion of a cylindrical shape that stretches infinitely. These steps aid in visualizing any cylindrical equation by recognizing the underlying 2D shape and determining the axis of extension, which provides clarity to the learner on how such surfaces occupy space in 3D. This approach not only facilitates the sketching of the surface but also deepens the understanding of spatial relationships in geometry.