Find a polynomial for the sum of the shaded areas of the figure.

We have a circle of radius 7, centered at the origin, so we need to find a parametrization for c as follows:

Part (a) 

As shown in figure 1,  c is inducing a counterclockwise orientation, so using trigonometry the equations are given by:



As:



if then and  that is the condition for the problem.

Part (b) 

As shown in figure 2,  c is inducing a clockwise orientation, so using trigonometry the equations are given by:



As:



if then and  that is the condition for the problem.

Part (c) 

As shown in figure 3,  c is now centered at the point (4, 3). Let's assume that it is inducing a counterclockwise orientation, so using trigonometry and doing a displacement of 4 in x-axis and 3 in y-axis, the equations are given by:



Where: