91Ó°ÊÓ

In geometry, the term **traces** refers to the intersection of a surface with particular planes. When we analyze a cylinder, like the one described by the equation \(x^{2}+y^{2}=4\), understanding its traces can help us comprehend its shape better.

For a cylinder extending along the z-axis, we primarily observe how it intersects with the standard coordinate planes, such as the xy-plane, yz-plane, and xz-plane.

• In the **xy-plane**, where \(z = 0\), the trace is a perfectly round circle. This is due to only the terms containing \(x\) and \(y\) being present in the equation.
• In the **yz-plane** and **xz-plane**, the trace is a straight line because the equation does not include \(z\). Thus, no restriction applies along these axes, making the cylinder extend infinitely along the z-axis.

By sketching these traces, we gain insight into the spatial presence and structure of the cylinder.

The equation \(x^{2}+y^{2}=4\) beautifully outlines a circle in the xy-plane.

This circle is centered at the origin \((0,0)\) with its equation derived from the standard form for a circle, which is \(x^2 + y^2 = r^2\). Here, the absence of z implies that the circle is perfectly flat and exists in the two-dimensional xy-plane.

The circle is a fundamental feature of many three-dimensional bodies. For the given exercise, sketching it involves drawing all points equidistant from the center, forming a closed loop in the xy-plane.
Understanding this plane aids in visualizing how the two-dimensional concept extends into three-dimensional space to form a cylinder.

To determine the **radius of the circle**, we revisit the circle's standard equation, \(x^{2}+y^{2}=r^{2}\).

In the given equation, the right-hand side equals 4, providing us a clue about the circle's size. The radius \(r\) is obtained by taking the square root of 4. Thus, we calculate:

• \(r = \sqrt{4}\)
• \(r = 2\)

This radius represents the distance from the circle's center, located at the origin, to any point on its circumference.

For students visualizing this, draw a circle with radius 2 in the xy-plane to represent the cross-section of the cylinder. This understanding of the radius not only defines the circle but directly influences the cylinder's shape, helping to piece together the whole three-dimensional picture.