Question: How to numerically integrate the numerical derivative of a numerical solution to a PDE?

I want to test how well Maple's numeric solver respects a variety of conservation laws. To do so, I must numerically integrate a numerical derivative; however, every method I've tried using does not work. In essence, I need the following code to work.

restart;
with(PDEtools); with(plots); with(DEtools);
pde := diff(v(x, t), t, t) = diff(v(x, t), x, x);
f := xi -> exp(-xi^2);
a := -10; b := 10; dx := 1/50; t_final := 10;

pds := pdsolve(pde, {v(a, t) = 0, v(b, t) = 0, v(x, 0) = f(x + 5), D[2](v)(x, 0) = -eval(diff(f(x), x), x = x + 5)}, numeric, range = a .. b, time = t, spacestep = dx);

sol_proc := rhs(pds:-value(output = listprocedure)[3]);
sol := (x, t) -> piecewise(a < x and x < b, sol_proc(x, t), 0);
int(fdiff(sol(x, t), [t = 0]), a .. b);

Error, (in depends) too many levels of recursion