What is faraday's law of electromagnetic induction in integral form?

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Faraday's law of induction makes use of the magnetic flux ΦB through a hypothetical surface Σ whose boundary is a wire loop. Since the wire loop may be moving, we write Σ(t) for the surface. The magnetic flux is defined by a surface integral:

where dA is an element of surface area of the moving surface Σ(t), B is the magnetic field, and B·dA is a vector dot product (the infinitesimal amount of magnetic flux). In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic flux lines that pass through the loop.

When the flux changes—because B changes, or because the wire loop is moved or deformed, or both—Faraday's law of induction says that the wire loop acquires an EMF , defined as the energy available per unit charge that travels once around the wire loop (the unit of EMF is the volt). Equivalently, it is the voltage that would be measured by cutting the wire to create an open circuit, and attaching a voltmeter to the leads. According to the Lorentz force law,

the EMF on a wire loop is:

where E is the electric field, B is the magnetic field (aka magnetic flux density, magnetic induction), dℓ is an infinitesimal arc length along the wire, and the line integral is evaluated along the wire (along the curve the conincident with the shape of the wire).

The EMF is also given by the rate of change of the magnetic flux:

where is the magnitude of the electromotive force (EMF) in volts and ΦB is the magnetic flux in webers. The direction of the electromotive force is given by Lenz's law.

For a tightly wound coil of wire, composed of N identical loops, each with the same ΦB, Faraday's law of induction states that where N is the number of turns of wire and ΦB is the magnetic flux in webers through a single loop.

the equations in this are in question form look for the equation in the source url's

Faraday's law of electromagnetic induction is the magnitude of induced EMF in circuit is equal to the time rate of change of magnetic flux trough the circuit.

Closed-path integral of E*ds = -d/dt where phi = mag. flux = B*A. A is the circular area of the solenoid. This integral applies both inside & outside the solenoid. Since flux is proportional to current,

-d/dt = constant.

So if we choose a closed path of integration of radius r outside the solenoid but concentric with it, E*ds integrated over this path = 2pi*r*E = constant, or E = /r. So the field is symmetrical about the center of the solenoid and diminishes with 1/r.