Channel flow Water flows in a shallow semicircular channel with inner and outer radii of 1 m and 2 m (see figure). At a point P(r,θ) in the channel, the flow is in the tangential direction (counterclock wise along circles), and it depends only on r, the distance from the center of the semicircles.


a. Express the region formed by the channel as a set in polar coordinates.

Verified step by step guidance

1. Recall that in polar coordinates, a point is represented as \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle measured from the positive x-axis.
2. The channel is a semicircular region bounded by two circles with inner radius 1 m and outer radius 2 m, and it is a semicircle, so the angle \(\theta\) will range over half of the full circle.
3. Since the channel is semicircular, \(\theta\) will vary from 0 to \(\pi\) (radians) to cover the upper half of the plane.
4. The radial coordinate \(r\) will vary between the inner radius and outer radius, so \(1 \leq r \leq 2\).
5. Therefore, the region formed by the channel in polar coordinates can be expressed as the set of points \((r, \theta)\) such that \(1 \leq r \leq 2\) and \(0 \leq \theta \leq \pi\).

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a radius and an angle, denoted as (r, θ). The radius r measures the distance from the origin, while θ is the angle from a reference direction, typically the positive x-axis. This system is especially useful for describing circular or radial regions.

Region Description in Polar Coordinates

Describing a region in polar coordinates involves specifying the ranges of r and θ that define the area. For a semicircular channel, the radius varies between inner and outer boundaries, and the angle covers the semicircle's angular span. This allows precise mathematical representation of complex shapes.

Geometry of Semicircular Channels

A semicircular channel is formed between two concentric semicircles with different radii. Understanding its geometry requires recognizing the inner and outer radii as boundaries and the angular limits that define the semicircle. This helps in setting up integrals or equations for flow analysis.