Question

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

Short answer

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

Step-by-step solution

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.

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\(\theta\).

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 \(\frac{\pi}{10}\).