Prove that u v2 u 2 v2 2u v

Chapter 10, Problem 83

(choose chapter or problem)

Spiral of Archimedes   The curve represented by the equation \(r=a \theta\), where a is a constant, is called the spiral of Archimedes.

(a) Use a graphing utility to graph \(r=\theta\), where \(\theta \geq 0\). What happens to the graph of \(r=a \theta\) as a increases? What happens if \(\theta \leq 0\)?

(b) Determine the points on the spiral \(r=a \theta(a>0, \theta \geq 0)\), where the curve crosses the polar axis.

(c) Find the length of \(r=\theta\) over the interval \(0 \leq \theta \leq 2 \pi\).

(d) Find the area under the curve \(r=\theta\) for \(0 \leq \theta \leq 2 \pi\).

Text Transcription:

r=a theta

r=theta

theta >= 0

theta <= 0

r=a theta(a>0, theta >= 0)

0 <= theta <= 2 pi