Problem 32 - Essential Calculus Early Transcendentals

Question

The solid cylinder that lies on or below the plane \(z=8\) and on or above the disk in the \(x y\) -plane with center the origin and radius 2

Step-by-Step Solution

Answer

The solid cylinder is defined by: \\(x^2 + y^2 \le 4\\) and \\(0 \le z \le 8\\).

Step 1: Identify the Three-Dimensional Space

The exercise describes a solid cylinder. This cylinder lies between two limits: a plane at \(z=8\) and a disk centered at the origin in the \(xy\)-plane with a radius of 2. We need to consider the shape and size of this cylinder.

Step 2: Recognize the Bounds

The cylinder described has a height extending from the \(xy\)-plane (\(z=0\)) to \(z=8\). Its base is a disk centered at the origin with radius extending from \((-2, -2)\) to \(2, 2\) in the \(xy\)-plane.

Step 3: Analyze the Cylinder's Base

The base of the cylinder is defined by the equation \(x^2 + y^2 \leq 4\), representing a disk with radius 2. This will constrain the values of \(x\) and \(y\) while \(z\) varies from 0 to 8.

Step 4: Formulate the Geometric Representation

Hence, the solid cylinder can be represented using inequalities. The bounds are \(x^2 + y^2 \leq 4\) and \0 \leq z \leq 8\. These describe the full spatial limits of the solid cylinder.

Key Concepts

Three-Dimensional Shapes

A three-dimensional shape is an object that has depth in addition to height and width. When we talk about three-dimensional shapes in geometry, we're usually discussing forms like cubes, spheres, and cylinders.

The solid cylinder in this problem is a classic example of a three-dimensional shape. It is defined by its circular base and consistent diameter extending through its height. The important parts of a cylinder include:

• Base: A circle at the bottom and top (if it's not designated as open or incomplete).
• Height: Represents the distance between the two bases of a cylinder or from the base to the upper surface if it’s considered closed at one end.
• Radius: The distance from the center of the circle to any point on its perimeter.

The cylinder discussed here has a base at the origin in the xy-plane extending to a radius of 2. Its height stretches vertically in the z-direction from 0 to 8.

Geometric Bounds

Geometric bounds define the limits within which a shape exists. These are crucial for understanding the space occupied by any three-dimensional object.

For the cylinder in this exercise, the geometric bounds include:

• Radius Bound: The base of the cylinder is a disk in the xy-plane centered at the origin. It includes all points where the equation \(x^2 + y^2 \leq 4\) holds true, limiting the radius to 2.
• Height Bound: The cylinder exists between the planes \(z=0\) and \(z=8\), introducing the vertical limit of the object in the 3D space.

These bounds create a confined region in space where the cylinder's solid form can be visualized. Understanding these constraints helps in visualizing and computing properties like volume and surface area.

Spatial Representation

Spatial representation involves visualizing an object in a three-dimensional coordinate system. This means imagining the object not just as a shape on paper but as a form existing in 3D space.

The cylinder in this exercise is imagined by starting with its spatial base description — the circular disk in the xy-plane — and extending upwards to a specified height. Here's how we break it down:

• Base representation: The base's circle is represented by the inequality \(x^2 + y^2 \leq 4\). This defines all the points within a circle of radius 2 centered at the origin.
• Vertical representation: Extending from \(z=0\) to \(z=8\) means every point on the circle has a vertically aligned counterpart reaching up to a height of 8.

Such a representation makes it easier to understand the shape's size and placement in a universal 3D space. Spatial representation is key for engineers, architects, and designers when working with complex 3D models and calculations.