Introduction:
In this problem, we are given a cardioid defined by the polar equation r = 1 - cos(θ) and a parabola defined by the polar equation r(1 - cos(θ)) = 1. We need to find the area enclosed by the cardioid and outside the parabola using double integration.

Step 1: Graphing the equations:
To visualize the problem, let's graph the cardioid and the parabola. We can plot points on a polar coordinate system by substituting different values of θ into the equations. By doing this, we can see the shape and the region we are interested in.

Step 2: Finding the points of intersection:
To find the area enclosed by the cardioid and outside the parabola, we need to find the points of intersection between the two curves. These points will determine the boundaries of the region we are interested in.

Step 3: Setting up the integral:
To find the area, we can use a double integral in polar coordinates. The general formula for finding the area enclosed by a polar curve is given by:

A = ½ ∫[a,b] (r(θ))^2 dθ

In this case, the limits of integration will be the angles at which the cardioid and the parabola intersect.

Step 4: Evaluating the integral:
We can now evaluate the integral using the limits of integration we found in the previous step. By substituting the polar equation of the cardioid into the integral formula, we can simplify the expression and calculate the area.

Step 5: Final answer:
After evaluating the integral, we will obtain a numerical value for the area. This will be the final answer to the problem, representing the area enclosed by the cardioid and outside the parabola.

Conclusion:
Using double integration, we can find the area lying inside the cardioid r = 1 - cos(θ) and outside the parabola r(1 - cos(θ)) = 1. By graphing the equations, finding the points of intersection, setting up and evaluating the integral, we can calculate the exact value of the enclosed area.