91Ó°ÊÓ

The equation given is a polar equation of the form \(r = a \cos(n\theta)\). Here, the polar equation \(r=3 \cos 2 \theta\) describes a four-leaved rose. This is because when \(n\) is even in \(a \cos(n\theta)\), the number of petals is \(2n\). For this equation, \(n = 2\) and the amplitude \(a = 3\). Hence, there are four leaves, each extending a distance of 3 units from the origin.

To sketch one leaf of the rose, we focus on the range of \(\theta\) that produces one petal. For \(r = 3 \cos 2 \theta\), one leaf is produced in the range when \(\theta\) varies from \(-\frac{\pi}{4}\) to \(\frac{\pi}{4}\). To graph this, plot \(r\) versus \(\theta\) within this range. Notice the petal begins at the origin, extends out to \(r = 3\), and returns to the origin.

The area \(A\) enclosed by one petal in polar coordinates is given by the integral:\[A = \frac{1}{2} \int_{\theta_1}^{\theta_2} r^2 \, d\theta\]Substitute \(r = 3 \cos 2 \theta\) and the limits \(\theta_1 = -\frac{\pi}{4}\) and \(\theta_2 = \frac{\pi}{4}\). This results in the integral:\[A = \frac{1}{2} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} (3 \cos 2 \theta)^2 \, d\theta\]

First, simplify \((3 \cos 2 \theta)^2 = 9 \cos^2 2\theta\). Thus, the integral becomes:\[A = \frac{9}{2} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos^2 2\theta \, d\theta\]Using the identity \(\cos^2 x = \frac{1 + \cos 2x}{2}\), this becomes:\[A = \frac{9}{2} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{1 + \cos 4\theta}{2} \, d\theta\]This further simplifies to:\[A = \frac{9}{4} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} (1 + \cos 4\theta) \, d\theta\]Now, evaluate:\[A = \frac{9}{4} \left[ \theta + \frac{1}{4} \sin 4\theta \right]_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\]After substituting the limits:\[A = \frac{9}{4} \left( \frac{\pi}{4} - \left(-\frac{\pi}{4}\right) + \frac{1}{4} \sin \pi - \frac{1}{4} \sin(-\pi) \right)\]This simplifies to:\[A = \frac{9}{4} \cdot \frac{\pi}{2} = \frac{9\pi}{8}\]

Each petal of the four-leaved rose has an area of \(\frac{9\pi}{8}\). Since we only calculated for one leaf, this is the total area for one petal.