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Chapter 9: Problem 8

Find the area of the region. One petal of \(r=\cos 5 \theta\)

Short Answer

Expert verified
The area of one petal of the rose curve \(r = cos(5\theta)\) is \(\frac{\pi}{10}\) square units.

Step by step solution

01

Understanding the Curve

In order to find the area of one petal of the rose curve, firstly understand its configuration. A rose curve defined by \(r = cos(n\theta)\), has n petals if n is odd.
02

Formulation of the Integral

The area A of a polar curve from \(\theta = a\) to \(\theta = b\) is given by \(\frac{1}{2}\) integral from a to b [\(r(\theta)\)^2] d\(\theta\). For one petal of \(r = cos(5\theta)\), \(a = 0\) and \(b = \frac{\pi}{5}\), hence the integral for the area would be \(\frac{1}{2}\)integral from 0 to \(\pi/5\) \((cos(5\theta))^2\) d\(\theta\).
03

Evaluating the Integral

The expression inside the integral, \((cos(5\theta))^2\) can be manipulated using power-reduction identity to become \(\frac{1}{2} + \frac{1}{2}cos(10\theta)\). Substituting this manipulation into the integral and evaluating it from 0 to \(\pi/5\), gives \(\frac{\pi}{10}\).

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