Question

Describe the curve, surface or region in space determined by the given bounds. Bounds in cylindrical coordinates: (a) \(r=1, \quad 0 \leq \theta \leq 2 \pi, \quad 0 \leq z \leq 1\) (b) \(1 \leq r \leq 2, \quad 0 \leq \theta \leq \pi, \quad 0 \leq z \leq 1\) Bounds in spherical coordinates: (c) \(\rho=3, \quad 0 \leq \theta \leq 2 \pi, \quad 0 \leq \varphi \leq \pi / 2\) (d) \(2 \leq \rho \leq 3, \quad 0 \leq \theta \leq 2 \pi, \quad 0 \leq \varphi \leq \pi\)

Short answer

(a) Cylinder of radius 1 and height 1; (b) Hemispherical cylindrical shell; (c) Upper hemisphere of radius 3; (d) Spherical shell from radius 2 to 3.

Step-by-step solution

Understanding the Cylindrical Bounds for Part (a)

The given bounds are in cylindrical coordinates: \(r=1,\ 0 \leq \theta \leq 2\pi,\ 0 \leq z \leq 1\). Here, the radius \(r\) is constant at 1, meaning geographically this describes a circular path in the \(xy\)-plane at a radius of 1. The angular range \(0 \leq \theta \leq 2\pi\) allows the circle to go a full 360 degrees, forming a full circular base. The elevation \(z\) ranges from 0 to 1, which lifts the circle into a cylinder that rises to height 1. Thus, these bounds outline the surface of a cylinder of radius 1 and height 1 with its base on the \(xy\)-plane.

Understanding the Cylindrical Bounds for Part (b)

For \(1 \leq r \leq 2,\ 0 \leq \theta \leq \pi,\ 0 \leq z \leq 1\), these bounds form a region in cylindrical coordinates. Here, \(r\) varies from 1 to 2, describing circular paths with radii from 1 to 2 centered at the origin. The angular bound \(0 \leq \theta \leq \pi\) limits this to the upper half-plane in the \(xy\)-plane. The \(z\) range from 0 to 1 places these circular paths vertically up, forming the shape of a cylindrical shell (annular region) with height 1, lying in the hemi-place above the \(xy\)-plane.

Understanding the Spherical Bounds for Part (c)

This part involves spherical coordinates: \(\rho=3,\ 0 \leq \theta \leq 2\pi,\ 0 \leq \varphi \leq \pi/2\). Here \(\rho\) being constant at 3 means a sphere of radius 3. The \(\theta\) range allows for full rotation around the \(z\)-axis. The polar angle \(\varphi\) from 0 to \(\pi/2\) limits this region to the upper hemisphere above the \(xy\)-axis (the northern hemisphere of the sphere). Thus, this describes the upper hemisphere of a sphere with a radius of 3.

Understanding the Spherical Bounds for Part (d)

The bounds given are \(2 \leq \rho \leq 3,\ 0 \leq \theta \leq 2\pi,\ 0 \leq \varphi \leq \pi\). Here, \(\rho\) values from 2 to 3 describe a spherical shell, a region bounded by spheres of radii 2 and 3. The theta coverage \(0 \leq \theta \leq 2\pi\) provides a full wrap-around. The angle \(\varphi\) from 0 to \(\pi\) outlines the whole range of vertical angles, from the north pole to the south pole, constructing a complete spherical shell between radii 2 and 3.

Key concepts

spherical coordinates

Spherical coordinates are a three-dimensional coordinate system that represent points by their distance from the origin, \(\rho\), the angle \(\theta\) from the positive x-axis, and the angle \(\varphi\) from the positive z-axis. This system is extremely useful for situations involving symmetry around a point, such as entire spheres or parts of spheres.
In the original exercise, we see a region defined by \(\rho=3\), \(0 \leq \theta \leq 2\pi\), and \(0 \leq \varphi \leq \pi/2\). Here, \(\rho=3\) indicates all points are on the surface of a sphere with a radius of 3 units. The range for \(\theta\) implies full rotation around the z-axis, and \(\varphi \leq \pi/2\) confines points to the upper hemisphere, which is the top half of the sphere. Such understanding of spherical coordinates helps identify regions like upper hemispheres or entire spheres.
Remembering this standard spherical system is essential for delving into more complex 3D shapes.

cylinder

A cylinder is a three-dimensional shape with two parallel circular bases and a curved surface connecting these bases. Cylindrical coordinates express points based on a radius \(r\), an angle \(\theta\) around the axis, and a height \(z\). These are particularly useful for describing objects with a circular symmetry about a central axis, like pipes or tanks.
In the provided exercise, one part described using cylindrical coordinates shows bounds of \(r=1\), \(0 \leq \theta \leq 2\pi\), and \(0 \leq z \leq 1\). This describes a cylinder where the radius \(r\) is constant at 1. The angular range completes a full circle, capturing a complete circular base. Elevating this base from \(z=0\) to \(z=1\), we identify the complete outline of a cylinder found in real-world objects like tin cans or pillars.
Understanding how cylindrical coordinates relate to cylinders helps bridge familiarity with real objects and mathematical models.

spherical shell

A spherical shell is a hollow region between two concentric spherical boundaries. Spherical shells are often used to visualize spaces like the atmosphere around a planet or the coating around particles.
Looking at the problem, the conditions \(2 \leq \rho \leq 3\), \(0 \leq \theta \leq 2\pi\), and \(0 \leq \varphi \leq \pi\) form a spherical shell. The variable \(\rho\) ranges from 2 to 3, and \(\theta\) extends completely around the z-axis while \(\varphi\) spans from the top to the bottom of the sphere. Essentially, it's a complete shell that envelopes full spheres of radius 2 and 3. This spatial region encases a volume in three-dimensional space like layers of an onion.
Grasping spherical shells in mathematics can simplify problems related to physics or other sciences that involve concentric layers.

radius

The radius in mathematics is a fundamental geometric concept that defines a line segment from the center of a circle or sphere to any point on its perimeter or surface. It plays a vital role in defining the size and shape of spherical and cylindrical objects.
In cylindrical and spherical coordinates, the radius is denoted differently--- just as \(r\) or \(\rho\) respectively. For cylindrical coordinates, \(r\) signifies the distance from the z-axis. Meanwhile, in spherical coordinates, \(\rho\) represents the distance from the origin to the point.
For example, in cylindrical problems like \(r=1\), this constant creates a circle of radius 1 in the xy-plane. Similarly, \(\rho=3\) in spherical coordinates describes a sphere's surface with a radius of 3. Being comfortable with the concept of radius across various systems is crucial for tackling geometry or calculus problems involving spatial understanding.