91Ó°ÊÓ

To find the area of the region bounded by the cardioid, we can use the formula $$ A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 d\theta, $$ where \(\alpha\) and \(\beta\) are the lower and upper bounds of the region in terms of the angle, \(\theta\). In this case, the angle \(\theta\) covers a range from \(0\) to \(2\pi\). So, we have $$ A = \frac{1}{2} \int_{0}^{2\pi} (3 - 3\cos\theta)^2 d\theta. $$ To solve this integral, we use trigonometric identities to simplify the expression. $$ A = \frac{1}{2} \int_{0}^{2\pi} (9 - 18\cos\theta + 9\cos^2\theta) d\theta. $$ Applying the trigonometric identity \(\cos^2\theta = \frac{1 + \cos(2\theta)}{2}\), $$ A = \frac{1}{2} \int_{0}^{2\pi} (9 - 18\cos\theta + \frac{9}{2} + \frac{9}{2}\cos(2\theta)) d\theta, $$ which further simplifies to $$ A = \frac{1}{2} \int_{0}^{2\pi} (9\cos(2\theta) - 18\cos\theta + 18) d\theta. $$ Now, integrate $$ A = \frac{1}{2} \left[-\frac{9\sin(2\theta)}{4} - 18\sin\theta + 18\theta\right]_{0}^{2\pi}. $$ Plug the limits and subtract $$ A = \frac{1}{2}(0) = 0, $$ and multiply by \(\frac{1}{2}\), $$ A = 0. $$ The total area of the region is \(0\). Now, let's find the centroid of the region.

To calculate the centroid, we will find \(x\) and \(y\) as follows: $$ x = \frac{1}{2A} \int_{0}^{2\pi} r^3 \cos\theta d\theta, $$ $$ y = \frac{1}{2A} \int_{0}^{2\pi} r^3 \sin\theta d\theta. $$ Plugging in the given curve \(r = 3 - 3\cos\theta\) and \(A = 0\), $$ x = \frac{1}{2(0)} \int_{0}^{2\pi} (3 - 3\cos\theta)^3 \cos\theta d\theta, $$ $$ y = \frac{1}{2(0)} \int_{0}^{2\pi} (3 - 3\cos\theta)^3 \sin\theta d\theta. $$ Notice that we have a division by zero in both equations, which means they are undefined. Since both coordinates of the centroid are undefined, we cannot determine the centroid of the given constant-density plane region. This is because the total area of the region is 0, which means the region has no mass. Thus, the concept of the centroid does not apply in this case.