There are a couple of ways to get integral forms of Maxwell's equations. Gauss' law can be applied to the equation ## \nabla \cdot E=\rho/\epsilon_o ## to give ## \int \nabla \cdot E \, d^3x= \int E \cdot \, dA=Q/\epsilon_o ## where ## Q ## is the total charge enclosed, and the flux of the electric field is over the surface of the enclosed volume. An alternative integral form can be written for this equation: ## E(x)=\int \frac{\rho(x')(x-x')}{4 \pi \epsilon_o |x-x'|^3} \, d^3x' ## . This second form is essentially Coulomb's law (inverse square law) for the electric charge distribution ## \rho(x) ##. ## \\ ## Gauss' law can also be employed on the equation ## \nabla \cdot B=0 ##, but usually this one is just used in a qualitative form where the lines of flux for the magnetic field ## B ## are said to be continuous. In integral form, it reads ## \int B \cdot \, dA=0 ##. ## \\ ## ## \\ ## For the curl equations, Stokes theorem (## \int \nabla \times E \cdot \, dA=\oint E \cdot \, ds ##), is usually employed, e.g. Faradays law becomes ##\varepsilon= \oint E \cdot \, ds=-\int (\frac{\partial{B}}{\partial{t}}) \cdot \, dA ##. ## \\ ## For the curl B, in the steady state, again Stokes law is often employed to give ## \oint B \cdot ds=\mu_o I ##. An alternative integral form does exist for this one also in the steady state which is the Biot-Savart Law: ## B(x)=\int \frac{\mu_o J(x') \times (x-x')}{4 \pi |x-x'|^3} \, d^3x' ##. For the non-steady state, I believe the solutions of Maxwell's equations are found by the Lienard-Wiechert method, but that is likely to be beyond the scope of what you are presently doing. ## \\ ## It is worth noting that the integral forms of these, which is sometimes treated in courses in vector calculus, are a little more complicated than simply going from differentiation to integration.