Calculation of i_enclosed

From Figure (1) we see that we get a current i up through our path each time another turn comes up through the path. On the left side of the coil the current goes down into the paper, but these downward currents lie outside our path and therefore are not included in our evaluation of i_enclosed. Only the positive upward currents count, and i_enclosed is simply i times the number of turns that go up through the path.
To calculate the number of turns in a height h of the coil, we note that if the coil has a length L and a total of N turns, then the number of turns per unit length n is given by

and in a height h there must be nh turns

With nh turns, each carrying a current i, going up through our path, we see that i_enclosed must be
i_enclosed = inh

Figure 1: Right hand rule for using Ampere's law. We define the positive direction around the path as the direction you curl the fingers of your right hand when the thumb is pointing in the direction of the current through the path. (As you see, we can come up with a right hand rule for almost anything.)