   Chapter 10.5, Problem 79E

Chapter
Section
Textbook Problem

# Spiral of Archimedes The curve represented by the equation r = a θ , where a is a constant is called the spiral of Archimedes.(a) Use a graphing utility to graph r = θ , where θ ≥ 0 . What happens to the graph of r = aθ as a increases? What happens if θ ≤ 0 ? (b) Determine the points on the spiral r = a θ ( a > 0 , θ ≥ 0 ) , where the curve crosses the polar axis.(c) Find the length of r = θ over the interval 0 ≤   θ ≤ 2 π .(d) Find the area under the curve r = θ for 0 ≤   θ ≤ 2 π .

(a)

To determine
What happens to graph of r=aθ as a increases and what happens if θ0.

Explanation

Given:

The curve represented by the equation r=aθ where a is constant is called spiral of archimedes

Explanation:

As a increases the spiral opens more rapidly

If θ<</

(b)

To determine

To Calculate: The points on the spiral r=aθ(a>0,θ0); where the curve crosses the polar axis

(c)

To determine

To Calculate: The Length of the curve r=θ over the interval 0θ2π.

(d)

To determine

To Calculate: The area under the curve of the polar equation r=θ over the interval 0θ2π

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