Something like the following, I suppose?
julia> using QuadGK, NonlinearSolve, Integrals
julia> f(y, p) = y^2 - p.b
f (generic function with 2 methods)
julia> g(x, p) = solve(IntegralProblem(f, (zero(x), x), p), QuadGKJL()).u - p.a
g (generic function with 1 method)
julia> p = (; rtol = 1e-2, a = 0.1, b = 0.5)
(rtol = 0.01, a = 0.1, b = 0.5)
julia> prob = IntervalNonlinearProblem(g, (-1.0, 1.0), p)
IntervalNonlinearProblem with uType nothing. In-place: false
Interval: (-1.0, 1.0)
julia> solve(prob)
retcode: Success
u: -0.2058119300266939
julia> g(ans.u, p)
0.0
julia> using GLMakie
julia> plot(-1:0.1:1, g.(-1:0.1:1, (p,)))
