Question

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

Short answer

The bounded region is the portion of the hemisphere defined by the equation \(z=\sqrt{4-x^{2}-y^{2}}\) that lies above the XY plane and beneath the plane \(y=2z\).

Step-by-step solution

Understand Equation 1

The equation \(z=\sqrt{4-x^{2}-y^{2}}\) is an upward opening hemisphere centered at origin with radius 2.

Understand Equation 2

The equation \(y=2z\) represents a plane that makes a right angle with the XZ plane. It passes through the origin and slopes upwards along the positive y-axis.

Understand Equation 3

The equation \(z=0\) is the XY plane, a flat horizontal plane at the base.

Visualize the bounded region

The region bounded by these equations is the portion of the hemisphere above the XY plane and beneath the plane \(y=2z\).

Key concepts

3D Calculus

Three-dimensional calculus, or 3D calculus, is an extension of calculus to three dimensions. This field of mathematics is concerned with surfaces, curves, volumes, and their relationships in space. When we study 3D calculus, we often analyze mathematical functions with variables for the three spatial dimensions—usually labeled x, y, and z.

In 3D calculus, you might encounter concepts such as partial derivatives, which represent the rate of change of a function in respect to one variable, while keeping the others constant. You'll also work with multiple integrals, which are used to calculate volumes or surface areas for complex shapes. Understanding these foundations is key to solving practical problems, such as finding the bounded region in a set of three-dimensional equations.

Hemisphere Equations

The equation of a hemisphere in three-dimensional space can be visualized as a half-sphere, resting on a plane. The standard form of a sphere centered at the origin with radius r is described by the equation \(x^2 + y^2 + z^2 = r^2\).

To represent a hemisphere, we consider only half of the sphere. For example, an upward opening hemisphere is bound by the XY plane. Such a hemisphere's equation typically includes a square root to account for the positive half, like the equation \(z = \sqrt{r^2 - x^2 - y^2}\), which represents the upper half of a sphere centered at the origin with radius r.

Plane Equations

Equations of planes are essential in 3D calculus for defining flat surfaces in space. A plane equation can be presented in various forms, such as standard, general, or point-normal form. The most basic plane through the origin is described by \(ax + by + cz = 0\), where a, b, and c are coefficients that determine the orientation of the plane.

In our exercise example, the plane equation \(y=2z\) implies a plane that intersects the z-axis, creating a tilted flat surface in the three-dimensional space. The slope and direction of the plane are essential when sketching and understanding the bounded regions created by their intersections with other surfaces.

Visualizing Three-Dimensional Space

Visualizing three-dimensional space is a skill that improves with practice and is crucial for solving problems in 3D calculus. It requires us to imagine how different 3D objects such as spheres, planes, and cylinders interact. For instance, think of a 3D coordinate system where the x, y, and z axes extend in the cardinal directions.

When visualizing the 3D space, use simple techniques like breaking down objects into familiar shapes or projecting them onto 2D planes to aid comprehension. For example, when sketching a bounded region, you might imagine slicing the 3D objects with the planes they're bound by, or use shading to emphasize the areas of interest. These methods help to clarify the region of interest defined by an intersection of surfaces or boundaries.