I am trying to find the area enclosed by 4 piecewise smooth curves. As can be seen from the figure,
BLACK curve is a segment of a circle, C
X = 25*cos(a) Y = 25*sin(a) -0.8622 <= a <= 2.3262
ORANGE curve is a segment of an ellipse,E1
X = 18*cos(a) + 0 Y = 55*sin(a) + 35 3.4520 <= a <= 4.3982
GREEN curve is a line,L
X = -5.5623 + a*(9.4362 -5.5623) Y = -17.3081 + a*(-2.6817-17.3081) 0 <= a <= 1
and RED curve is a segment of another ellipse E2
X = 20*cos(a) + 25 Y = 35*sin(a) + 35 3.8206 <= a <= 4.2607
I know all the parametric equations for the curves and the parameters of each vertex with respect to the curve equations.
For example, the vertice at (-17.14,-18.2) has the parameter of 2.3262, in the range of [0, 2*pi], with respect to C.
To find the area enclosed by these curves, I tried to use the brute force way, that is, to integrate each curve segment equations. It works fine for the segment of the curve where X - Y values are bijective. Otherwise, the integration of some part of the curve will cancel out some other part of the curve. An example is the BLACK curve.
To my understand, Green's theorem can solve this. But how do I setup the curves to use Green's theorem? And also, Is there other ways to find the area enclosed by these curves conveniently?