...an integral with a variable lower limit...

Are you trying to plot $y=f(b) - \int_x^b f(t)\;dt$ for the some function $f$ over some interval $a \le x \le b$? Or maybe it's plus, $y=f(b) + \int_x^b f(t)\;dt$? The initial value $f(b)$ seems odd, but it follows from the graphs intersecting at $x=b=\pi$, assuming one curve is the integral and the other is the integrand.

1. I wonder what you're trying to plot? Can you give a formula such as I gave? Which curve is the integral and which is the integrand?

2. I doubt the first graph is the correct one, if one of the curves is the integral of the other. There should be a vertical shift, unless by random coincidence the shift is zero. If the graphs are phase-shifted sine curves, then the first graph is wrong. The following gives the vertical shift for $y=f(a) + \int_a^x f(t)\;dt$ where $f$ is a phase-shifted sine:

   Integrate[
     a*Sin[-Pi + b] + Integrate[a*Sin[t + b], {t, -Pi, x}],
     {x, -Pi, Pi}]/(2 Pi)
   (*  -a (Cos[b] + Sin[b])  *)
   

So the vertical shift is zero exactly when $\tan b = -1$, which is not the case in the first graph (assuming they are sines).