Question

Find the area of the regions bounded by the following curves. The complete three-leaf rose \(r=2 \cos 3 \theta\)

Short answer

Answer: The area of the regions bounded by the curve is \(\pi\).

Step-by-step solution

1. Find the Limits of Integration

To find the limits of integration for \(\theta\), we need to find the points where \(r=0\). To do this, solve the equation \(r=2\cos(3\theta)=0\) for \(\theta\). $$ 0 = 2\cos(3\theta) \\ \cos(3\theta) = 0 $$ As we know, if \(\cos(x) = 0\), then \(x = \frac{(2n+1)\pi}{2}\), where \(n\) is an integer. Therefore, $$ 3\theta = \frac{(2n+1)\pi}{2} \\ \theta = \frac{(2n+1)\pi}{6} $$ For the first petal, we can use \(n=0\) and \(n=1\). So, the limits of integration are \(\theta = 0\) and \(\theta = \frac{\pi}{3}\).

2. Calculate the Area of One Petal

To calculate the area of one petal, we will use the following polar coordinate area formula: $$ A_\text{petal} = \frac{1}{2}\int_{\theta_1}^{\theta_2} r^2 d\theta $$ We have the limits of integration: \(\theta_1=0\) and \(\theta_2=\frac{\pi}{3}\), and the function \(r=2\cos(3\theta)\). Thus, the integral is: $$ A_\text{petal} = \frac{1}{2}\int_{0}^{\frac{\pi}{3}} (2\cos(3\theta))^2 d\theta $$

3. Calculate the Integral

Now, we'll calculate the integral: $$ A_\text{petal} = \frac{1}{2}\int_{0}^{\frac{\pi}{3}} 4\cos^2(3\theta) d\theta $$ Using the double angle formula, we can rewrite \(\cos^2(3\theta)\) as \(\frac{1+\cos(6\theta)}{2}\): $$ A_\text{petal} = \frac{1}{2}\int_{0}^{\frac{\pi}{3}} 4\left(\frac{1+\cos(6\theta)}{2}\right) d\theta $$ Now, we simplify and evaluate the integral: $$ A_\text{petal} = \frac{1}{2}\int_{0}^{\frac{\pi}{3}} (2+2\cos(6\theta)) d\theta \\ A_\text{petal} = \frac{1}{2}\left[2\theta + \frac{\sin(6\theta)}{3}\right]_0^\frac{\pi}{3} $$

4. Evaluate the Integral and Calculate the Total Area

By plugging in the limits of integration, we can find the area of one petal: $$ A_\text{petal} = \frac{1}{2}\left[2\left(\frac{\pi}{3}\right) + \frac{\sin(2\pi)}{3} - \left(2(0) + \frac{\sin(0)}{3}\right)\right] \\ A_\text{petal} = \frac{1}{2}\left[\frac{2\pi}{3}\right] \\ A_\text{petal} = \frac{\pi}{3} $$ As there are 3 petals, we multiply the area of one petal by 3: $$ A_\text{total} = 3 \times A_\text{petal} \\ A_\text{total} = 3 \times \frac{\pi}{3} \\ A_\text{total} = \pi $$ So, the area of the regions bounded by the curve \(r=2\cos3\theta\) is \(\pi\).