Centroid of the upper half of cardioid curve

Question

The cardioid is given by polar coordinates: $$C: r=\frac{1+\cos(\phi)}{2}, \quad \phi \in(0; \pi)$$

I established that the lenght of the curve is equal to $2$, but I'm having great difficulties with integrals:

$$\int_C x \, dC$$ and $$\int_C y \, dC$$

Do You have any suggestions regarding the solution, or better yet the solution itself?

Answer 1

Since the cardioid is more natural in polar coordinates, we can directly measure the arc length in polar coordinates via $$ds=\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}dr$$ as derived here. So we have $$\bar{x} \int_C dC = \int_C x \ dC = \int_0^\pi r\cos(\theta)\sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}dr$$ This integral is fairly doable and WolframAlpha can help if you get stuck. The situation for $y$ is nearly identical.