Question

Find the area of the region inside one leat of the three-leaved rose r=4 cos 30 . The area of the region is (Type an exact answer, using π as needed.)

Answer 1

To find the area of one leaf of the three-leaved rose with polar equation r=4 cos(3\theta), integrate \(\frac{1}{2}(r(\theta))^2\) from 0 to \(\frac{\pi}{3}\) and express the result using \(\pi\) as needed.

Step-by-step explanation:

The student's question pertains to finding the area of one leaf of the three-leaved rose represented by the polar equation r=4 cos(3\theta). To compute the area of one leaf of the rose, we need to use the formula for the area in polar coordinates, which is \(A = \frac{1}{2} \int (r(\theta))^2 d\theta\). The three-leaved rose will complete one full leaf as \(\theta\) goes from 0 to \(\frac{\pi}{3}\), since the full rose has three leaves and symmetry over the interval \([0, 2\pi]\).

The area of one leaf can be calculated as follows:

1. Set up the integral: A = \frac{1}{2}\int_0^{\frac{\pi}{3}} (4 \cos(3\theta))^2 d\theta.
2. Compute the integral using trigonometric identities and integration techniques.
3. Simplify the result to get the area in terms of \(\pi\).

The exact computation would involve evaluating the definite integral of the squared polar equation within the specified limits.