I do not really what you means by an arch. I take its architectural sense
So I imagine that you want to know the volume under an arch of a certain lenght like the volumen of a vault, if that is your case, you must find the area under or inside the arch as a function of its lenght "x" let us call that area A(x)and then integregate A(x)dx between 0 and L, where L stands for the lenght of the vault.
The above it is assuming that the arch is changing its shape so A(x) is variable. If A is constant, now what you have to find is A and it depends of the shape of the arch, it could almost inmmediate if the shape is an easy one: any combination of circular, eliptical, parabolic or other integrable function what it is the usual case in architecture.
If it is not you must resort on numerical method like, Simpsom or Trapezoidal rules or other most sofisticated like Newton-Cotes, Tchebicheff,... not a big deal anyway.
If you give more details I may help you.