Calculate area enclosed by 4 curves

Question

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?

Thanks

Answer 1

I would add a new straight line segment, colored blue for purposes of naming, from the blue vertex, through the green vertex and extending to the black circle and add a new blue vertex at that intersection. This new blue segment is a chord of your black circle. I would also add straight arcs joining the black and brown vertices and the green and red vertices. You now have as diagram with 5 distinct areas. Two of these are triangles and one of is a segment of a circle — each of these is easy to compute. The remaining two areas can easily be turned into basic integrals (after rotating) which can be computed either by hand, if the parametric formulas are particularly nice, or numerically. Edit: since you’ve now added the parameterizations for the curves and noted they are all segments of ellipses it will be straightforward for you to compute the 5 areas mentioned above by hand. Hint: put each ellipse segment in standard position.