Berechnen Sie den Schwerpunkt der von der Kardiode \(r(\varphi)=1+\cos \varphi\), \(0 \leqslant \varphi<2 \pi\) begrenzten Fläche.

Polar coordinates are a two-dimensional coordinate system where each point is determined by a distance from a reference point (called the origin) and an angle from a reference direction. This system is especially useful for problems involving symmetry and periodic functions.
In the case of the cardiode, the polar equation is given by \(r(\varphi)=1+\cos\varphi\), and the angle, \(\varphi\), ranges from 0 to 2\pi. Polar coordinates simplify the description and calculation of areas for many curves.
To convert Cartesian coordinates (x, y) to polar coordinates (r, \(\varphi\)), we use the formulas:

• \(x = r \cos\varphi\)
• \(y = r \sin\varphi\)

Given this conversion, integrating in polar coordinates often helps simplify mathematical processes, as seen in this exercise where the cardioid's properties are more straightforward to analyze.

Integral calculus is a key concept in dealing with areas under curves, among other things. In this exercise, we use integral calculus to calculate the area enclosed by the cardioid and to find the centroid.
The general formula for the area enclosed by a polar curve is:

• \(A = \frac{1}{2} \int_{0}^{2\pi} r^2(\varphi) \, d\varphi\)

For the given cardioid, \(r(\varphi) = 1 + \cos\varphi\), the integrand becomes:

• \(A = \frac{1}{2} \int_{0}^{2\pi} (1+\cos\varphi)^2 \, d\varphi\)

Using integral calculus, we then break down the integral and solve it term-by-term. This method requires using trigonometric identities and understanding integral properties. Each part of the integrand can be integrated separately and then combined to find the total area.
This same process applies when identifying the centroid coordinates, involving more complex integrals that use the properties of symmetry and periodicity.

The calculation of the area enclosed by a curve is crucial in geometry and many applied fields. In this exercise, we find the area of the cardioid using polar coordinates.
First, we set up the integral for the area:

• \(A = \frac{1}{2} \int_{0}^{2\pi} (1+\cos\varphi)^2 \, d\varphi\)

Next, we simplify the integrand by expanding it:

• \((1+\cos\varphi)^2 = 1 + 2\cos\varphi + \cos^2\varphi\)

We then split the integral into manageable parts to integrate each term separately. Recognizing certain standard integrals and properties of trigonometric functions helps greatly:

• The integral of 1 from 0 to 2\pi is 2\pi
• The integral of \(\cos\varphi\) over one full period is zero
• The integral of \(\cos^2\varphi\) uses the identity \(\cos^2\varphi = \frac{1}{2}(1 + \cos(2\varphi)) \)

By combining these results, we get the total area:

• \(A = \frac{1}{2} (2\pi + \pi) = \frac{3\pi}{2}\)

Without this step-by-step integration process, calculating the area would be significantly more complex.