What Does the Area in Faraday's Law Represent?

[darksyesider]

In the formula: $emf = -\dfrac{d\phi}{dt}$, $\phi = \int B\cdot dA$.
Does the area mean only the part of the region which has a magnetic field (if it's constant)?

for example, if you have a loop around a toroid, and want to calculate the induced emf, would $\phi = B(\pi radiusoftoroid)^2$ or $\phi = B (\pi radiusofregion)^2$ ?

I personally believe the former, but cannot verify this.

 

[dauto]

The area is the area bounded by the loop along which the emf is being measured