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Chapter 12: Problem 39

Sketch the surfaces. $$x^{2}+z^{2}=1$$

Short Answer

Expert verified
It's an infinite cylinder oriented along the y-axis, with a circular cross-section of radius 1 in the xz-plane.

Step by step solution

01

Identify the Type of Surface

The given equation is \(x^2 + z^2 = 1\). This is a standard equation of a cylinder in three-dimensional space where the axis of the cylinder is along the y-axis.
02

Analyze the Cross-Sections

For a fixed value of \(y\), the cross-section of the surface is a circle in the \(xz\)-plane with radius 1. This is because \(x^2 + z^2 = 1\) is a circle centered at the origin with radius 1.
03

Describe the Surface

Since the equation does not involve \(y\), the circle is constant along the entire \(y\)-axis, forming an infinite cylinder that extends infinitely in the positive and negative y-directions. The circular base has a radius of 1.
04

Sketch the Surface

Draw the \(xz\)-plane and sketch a circle centered at the origin with radius 1. Then, along the \(y\)-axis, extend this circle as a cylinder. Ensure to convey that the cylinder stretches infinitely along the \(y\)-direction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Three-Dimensional Geometry
Three-dimensional geometry takes our understanding of shapes into a space where they have length, width, and height. In this context, we can explore more complex surfaces like planes, spheres, and cylinders. In three-dimensional geometry, each point is defined by three coordinates: usually denoted as
  • \(x\)
  • \(y\)
  • \(z\)
This allows us to represent spatial objects or surfaces using equations involving these variables, such as the equation for a cylinder. By understanding three-dimensional geometry, we can effectively interpret and sketch complex shapes in space, aided by visualizing how each point relates to the others in all three axes.
Cross-Sections
Cross-sections are slices of three-dimensional objects, and they help us understand the shape better by examining its 2D slices. For a cylinder, a cross-section taken parallel to its base and perpendicular to its height is a circle. In our example, the equation \(x^2 + z^2 = 1\) without a \(y\) component hints that cross-sections parallel to the base would remain consistent
  • These cross-sections exist along the \(xz\)-plane.
  • Each cross-section of the cylinder at a fixed \(y\) value forms a circle.
  • All circles have the same radius, which is 1 as denoted by the given equation.
Understanding these cross-sections is crucial as they reveal how the volume of the shape extends along one dimension, helping us visualize and sketch the overall surface.
Cylinder Sketching
Sketching a cylinder is a helpful way to visualize how it exists in three dimensions. In our case, we begin by considering the equation \(x^2 + z^2 = 1\), which tells us that at any point along the \(y\)-axis the cross-section in the \(xz\)-plane is a circle with radius 1.
  • Start by drawing a circle with radius 1 in the \(xz\)-plane. This represents a cross-section at \(y = 0\).
  • Next, extend this circle parallel to the \(y\)-axis to represent the continuous nature of the cylindrical surface.
  • Finally, show that the cylinder extends infinitely in both directions along the \(y\)-axis by drawing dashed lines or arrows.
A well-done sketch will help illustrate the infinite length and consistent circular cross-sections of the cylinder, reinforcing how it is represented in three-dimensional geometry.

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Most popular questions from this chapter

Find the distance between points \(P_{1}\) and \(P_{2}\). $$P_{1}(5,3,-2), \quad P_{2}(0,0,0)$$

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