Question

In your own words, describe \(r\) -simple regions and \(\theta\) -simple regions.

Short answer

An \(r\)-simple region in polar coordinates is defined by two functions of \(\theta\) and \(r\) being in an interval, whereas a \(\theta\)-simple region is defined by two functions of \(r\) and \(r\) being in an interval. Both regions are characterized by the way lines or circles intersect the region.

Step-by-step solution

Defining \(r\)-simple region

An \(r\)-simple region is a region in the polar coordinate system where, for \(r\) between two continuous functions of \(\theta\), and \(\theta\) between a closed interval, every radial line crossing through the region does so exactly once. In mathematical terms, an \(r\)-simple region is described as \(\{ (r, \theta) | \alpha(\theta) \leq r \leq \beta(\theta), \, a \leq \theta \leq b \}\) where \( \alpha(\theta) \) and \( \beta(\theta) \) are continuous functions of \(\theta\) and [a,b] is a closed interval.

Defining \(\theta\)-simple region

A \(\theta\)-simple region, on the other hand, is a region in the polar coordinate system for which, for \(\theta\) between two continuous functions of \(r\), and \(r\) between an interval, every circle centered at the origin and intersecting the region does so in a continuous arc or point. Formally, a \(\theta\)-simple region is described as \(\{ (r, \theta) | a \leq r \leq b, \, \alpha(r) \leq \theta \leq \beta(r) \}\) where \( \alpha(r) \) and \( \beta(r) \) are continuous functions of \(r\) and [a,b] is an interval.