Find the area of the region. one petal of $r=6 \cos (5 \theta)$

Step 1 ：The given equation is in polar coordinates. The area of a polar curve is given by the formula \(\frac{1}{2}\int_{a}^{b} r^2 d\theta\). Here, \(r = 6\cos(5\theta)\) is the equation of a rose curve with 5 petals. We are asked to find the area of one petal.

Step 2 ：The limits of the integral should be chosen such that they cover exactly one petal. For a rose curve with \(n\) petals described by \(r = a\cos(n\theta)\) or \(r = a\sin(n\theta)\), one petal is traced out as \(\theta\) ranges from \(0\) to \(\frac{2\pi}{n}\).

Step 3 ：In this case, \(n = 5\), so one petal is traced out as \(\theta\) ranges from \(0\) to \(\frac{2\pi}{5}\).

Step 4 ：So, we need to evaluate the integral \(\frac{1}{2}\int_{0}^{\frac{2\pi}{5}} (6\cos(5\theta))^2 d\theta\).

Step 5 ：The final answer is \(\boxed{3.6\pi}\).