In order to find the area of the region cut by the cardioid $r = 1 - \cos(\theta)$ in the second quadrant (meaning $\theta$ ranges from $\pi$ to $2\pi$), we can use the formula for the individual area in polar coordinates:
$A = 0.5 r^2 d\theta$.
To find the total area, we integrate this expression over the appropriate ranges. In this case, $\theta$ ranges from $\pi$ to $2 \pi$, and $r$ ranges from $0$ to $1 - \cos(\theta)$ (i.e., the equation of the cardioid).
So, the integral for the area A is:
$A = 0.5 \int_0^{1 - \cos(\theta)} \int_\pi^{2\pi} r^2 d\theta dr$
We integrate with respect to $r$ first, then with respect to $\theta$.
First, integrating with respect to $r$ gives:
$A = 0.5 \int_\pi^{2\pi} [\frac{1}{3} r^3]_0^{1 - \cos(\theta)} d\theta$
Simplify this expression to:
$A = 0.5 \int_\pi^{2\pi} \frac{1}{3} (1 - \cos(\theta))^3 d\theta$
Now, it is not easy to compute this integral analytically because it involves an integral of a cubic cosine function from $\pi$ to $2\pi$.
In general, one could use a numerical method such as Riemann sums, a trapezoidal rule, or Simpson's rule to approximate the definite integral. These methods involve approximating the area under the curve using simple geometric figures (rectangles, trapezoids, parabolas) and then summing the areas of these shapes to get the total area.
Notice that due to the complexity of the integral part, it is recommended to use software that is capable of performing definite integration. The solution would depend on the numerical approximation method used.