Problem 11 - Thomas Calculus

Question

In Exercises \(1-12,\) sketch the graph described by the following cylindrical coordinates in three-dimensional space. $$ 0 \leq r \leq 4 \cos \theta, \quad 0 \leq \theta \leq \frac{\pi}{2}, \quad 0 \leq z \leq 5 $$

Step-by-Step Solution

Answer

The problem describes a quarter-cylinder extending along the positive x-direction with height 5.

Step 1: Understand Cylindrical Coordinates

Cylindrical coordinates are in the form \((r, \theta, z)\). Here, \(r\) is the radius from the z-axis, \(\theta\) is the angle in the xy-plane from the positive x-axis, and \(z\) is the height above the xy-plane.

Step 2: Interpret Given Boundaries

We have three inequalities: 1. \(0 \leq r \leq 4 \cos \theta\) describes the radial distance as bounded by a function of \(\theta\).2. \(0 \leq \theta \leq \frac{\pi}{2}\) suggests the angle is confined to the first quadrant in the xy-plane.3. \(0 \leq z \leq 5\) limits the height from the xy-plane up to z = 5.

Step 3: Analyze Radial Boundary

The radial boundary \(r \leq 4 \cos \theta\) describes a variable boundary that changes with \(\theta\). For each \(\theta\), \(r\) is limited by \(4 \cos \theta\). At \(\theta = 0\), \(r\) can go up to 4, while at \(\theta = \frac{\pi}{2}\), \(r\) is limited to 0.

Step 4: Determine Region in xy-plane

In the xy-plane, when \(\theta\) varies from \(0\) to \(\frac{\pi}{2}\), and \(r\) varies from \(0\) to \(4 \cos \theta\), the region described is like a quadrant of a circle, compressed along the curve \(r = 4 \cos \theta\).

Step 5: Extend Region into 3D Space

Within each region in the xy-plane bounded as described, the value of \(z\) can range from 0 to 5. So, picture a quarter-cylinder shape opening towards the positive x-direction with a height of 5.

Step 6: Sketch the Graph

In your sketch: - Draw the xy-plane section, a circle quarter (radius height along x: 4, radius 0 at y-axis). - Extend this as a quarter-cylinder vertically up to z=5. It opens toward the positive x-axis and extends as a flat surface along the z-direction.

Key Concepts

3D Graph Sketching

When visualizing functions or equations in three-dimensional space, having a clear strategy for sketching graphs is key. In this exercise, we're tasked with sketching a shape defined by cylindrical coordinates.
To begin sketching, identify the boundaries of each coordinate. In this example, the radial coordinate

• **r** is bounded by a trigonometric function, giving us a circular shape constraint.
• **θ** is limited from 0 to π/2, restricting the shape to the first quadrant of the xy-plane.
• **z** varies from 0 to 5, determining the height of the 3D shape.

First, focus on draft drawing in the xy-plane, taking note of the radial and angular boundaries. Then, extend this projected shape into the z-dimension for a complete three-dimensional sketch. Remember, sketching in 3D often involves perceiving the shape from different angles to understand its structure and spatial relationships.

Cylindrical Coordinate System

The cylindrical coordinate system is a way to describe points in three-dimensional space using three parameters:

• **r**: The radial distance from the z-axis.
• **θ**: The angular coordinate, measuring the angle from the positive x-axis within the xy-plane.
• **z**: The height above the xy-plane, analogous to the z-value in Cartesian coordinates.

This system is particularly useful for objects with circular features. When given cylindrical coordinates, the radial boundary often changes with the angle, creating diverse shapes.
It's advantageous over Cartesian coordinates for problems involving symmetry about an axis, such as cylinders or spheres. Learning to interpret these parameters helps in identifying and visualizing different shapes they describe.

Radial Boundary Analysis

Radial boundary analysis involves understanding how the radius of a shape changes in relation to its angular position.
In our case, the radial boundary is described by the function \( r \leq 4 \cos \theta \). This particular setup shows how the maximum radius varies with angle:

• When \( \theta = 0 \), the maximum radius \( r \) extends up to 4.
• As \( \theta \) approaches \( \frac{\pi}{2} \), \( r \) diminishes to 0.

This forms a decreasing radial limit as the angle increases, shaping a quarter of a circle within the plane. Understanding these variations is crucial when sketching shapes with angular constraints.

3D Space Graph Representation

The journey from interpreting coordinates to representing them in 3D space involves extending the information between different planes. In this problem, after determining the region within the xy-plane:

• We notice it is a partial circle extending from high values along the x-axis, diminishing towards the y-axis.
• Then, based on the boundary for **z**, namely \( 0 \leq z \leq 5 \), the shape extends vertically.

This forms a quarter-cylinder extending upwards, with its open side facing along the positive x-axis.
By "stacking" layers from the xy-plane upwards to the maximum z-value, you can form a comprehensive picture of the described three-dimensional space. Graph representation in 3D becomes a scientific way to visualize and understand complex geometric constructs in tangible, manageable portions.