Numerical solution to integration of function which parameters define the integral limits

Question

I have following equation:

$\int^{\delta_c}_{\delta_c-\pi}\sqrt{h^2_c(\cos(\delta_c)^2-\cos(\delta)^2+\Omega_c(\delta_c-\delta)}\text{d}\delta=\pi$,
where $\delta_c=\frac{1}{2}\arcsin\left(\frac{\Omega_c}{h^2_c}\right)$ and I want to make a $\Omega_c(h_c)$ plot.

I have tried to solve it via NIntegrate and NSolve.

function[a_, b_, \[Delta]_] := NIntegrate[Sqrt[b(Cos[1/2 ArcSin[a/b]]^2 - Cos[\[Delta]]^2) + a (1/2 ArcSin[a/b] - \[Delta])], {\[Delta], 1/2 ArcSin[a/b] - \[Pi], 1/2 ArcSin[a/b]}]

NSolve[function[a, 0.1, \[Delta]] == \[Pi], a, Reals]

Answer 1

function[Ω_?NumericQ, h_?NumericQ] := 
NIntegrate[Sqrt[h^2 (Cos[1/2 ArcSin[Ω/h^2]]^2 - Cos[δ]^2) + Ω *(1/2 ArcSin[Ω/h^2] - δ)],
{δ, 1/2 ArcSin[Ω/h^2] - π, 1/2 ArcSin[Ω/h^2]}, 
Method -> "AdaptiveQuasiMonteCarlo"];

I used Method -> "AdaptiveQuasiMonteCarlo" in NDSolve,because is fast.

ContourPlot[function[Ω, h] == Pi, {h, -2, 2}, {Ω, -1/10, 1}, FrameLabel -> Automatic, 
PlotPoints -> 10] // Quiet