Question

How to write integral to calculate the area where a field is between two limits?

Answer 1

I would like to express the following quantity in a mathematical form, but cannot think how I would write the integral.  "The area on the surface of a cylinder where the magnetic field is between two limits, say, -∞ and -15 nt"  I attempted to write  ∫θ(B)=−15nT−∞∫z(B)=−15nT−∞B⃗ ⋅dθdz=A but I'm not too sure on the limit values. As you can see from the figure, the shape is curved, and doesn't have sides with constant values of θ or z, so I'm not certain how one would express the integral limits.  Top left: plot of the magnetic field on the surface  Top right: Field between two limits. It is the area of this shape that I want to express mathematically

Answer 2

we can use Green's theorem only if there happens to be a vector field F(x,y)F(x,y) so that 

f(x,y)=∂F2∂x−∂F1∂y.