Question

In Exercises \(27-30,\) sketch the region bounded by the graphs of the equations. $$ z=\sqrt{4-x^{2}}, y=\sqrt{4-x^{2}}, x=0, y=0, z=0 $$

Short answer

The region bounded by the given equations forms a hemisphere of radius 2 in the first octant.

Step-by-step solution

Understand the Equations

Firstly, observe the equations given. The equations \(x = 0, y = 0,\) and \(z = 0\) are the three coordinate planes in 3-dimensional space.

Equation Transformation

For a better understanding of the curves \(z=\sqrt{4-x^{2}}\) and \(y=\sqrt{4-x^{2}}\), rewrite them as \(z^{2}=4-x^{2}\) and \(y^{2}=4-x^{2}\). This form shows that they are semicircles in their respective planes, each with radius 2.

Visualize the Region

From this realization, it's clear that the two semicircles, in combination with the three planes, form a half-dome or a hemisphere shape. The straight edges will be on the three planes specified, while the curved face is dictated by the equation \(z=\sqrt{4-x^{2}}\) and \(y=\sqrt{4-x^{2}}\).

Sketch the Region

The sketch will be a hemisphere resting on the xy plane (z = 0), centered at the origin and a radius of 2 units. The curved surface lies in the first octant because all the coordinates are positive.

Key concepts

Understanding the Three-Dimensional Coordinate Planes

In the realm of three-dimensional geometry, the coordinate planes serve as the foundation for visualizing and solving problems. These planes are essentially flat, two-dimensional surfaces that divide space into different sections. There are three primary planes: the xy-plane, yz-plane, and zx-plane, each named after the axes it contains. When you have an equation like x = 0, this represents the yz-plane since it contains all points where the x-coordinate is zero, similarly y = 0 refers to the zx-plane, and z = 0 denotes the xy-plane.

The intersection of these planes at the origin (0,0,0) creates eight different sections called octants, similar to how quadrants divide a two-dimensional plane. When sketching regions in 3D space, it is important to consider which octants are included in the region defined by your equations. This understanding is crucial before moving on to more complex operations such as equation transformation and ultimately, sketching the defined regions.

The Process of Equation Transformation

Transforming equations is an indispensable tool in visualizing complex regions in 3D space. The transition from the given equations z=\(\sqrt{4-x^{2}}\) and y=\(\sqrt{4-x^{2}}\) to z^{2}=4-x^{2} and y^{2}=4-x^{2} makes it easier to see that these represent circles, or more specifically, semicircles, when restricted to positive y and z values. In this case, the radius of the semicircles is readily identified as 2, since a circle with radius r has the equation x^{2}+y^{2} = r^{2}.

This transformation is particularly useful because three-dimensional sketches can become very confusing without a clear understanding of the shapes involved. Rewriting equations in a familiar form allows one to exploit known geometrical facts to simplify the visualization process.

Sketching a Hemisphere in 3D Space

Sketching three-dimensional objects, such as a hemisphere, begins by understanding its basic shape and boundaries. A hemisphere is a half of a sphere, cut by a plane passing through the center of the sphere. In this exercise, the graphs of z=\(\sqrt{4-x^{2}}\) and y=\(\sqrt{4-x^{2}}\), combined with the planes at x=0, y=0, and z=0, define such a hemisphere.

When sketching, start by drawing the coordinate planes and then the semicircles in the yz and zx planes. Finally, connect these semicircles smoothly to form the curved surface of the hemisphere, which should appear to 'rise' from the xy plane. Ensuring that the sketch adheres to the equations' constraints—an origin-centered hemisphere resting on the xy plane with a radius of 2—is crucial for an accurate representation. This visual aid assists in comprehending the spatial relationships between the object and the coordinate planes.