My question is finding the total area covered by curves, such as the total area every curve in the following picture covers (from 100 on y axis to 200 on x axis):
In my case, the curves are parabolas of trajectories described by $y = \frac{1}{2}gt^2 + v_i \sin( \theta)t + y_0 $
This has probably been answered before, but I cannot find it. If there were 2 curves, the answer would be easy: just sum two areas and subtract the overlap.
On
To clarify Nitin's comment, suppose that $f_\theta(x)$ denotes the curve with parameter ("angle") $\theta$ (in radians). Supposing that your variable $x$ never exceeds $200$, you want
$$\int_0^{200} \sup_{\theta\in[0,\pi/2]} f_\theta(x) dx$$
This of course depends on the equations $f_\theta(x)$.
In example 5 of https://en.wikipedia.org/wiki/Envelope_(mathematics), it is shown that the envelope of a family of parabolas with constant initial velocity and different initial angles is also a parabola.
The equation is derived nicely.