Theorem 5.9: The length of the arc of the unit circle above the points a and c on the x axis is given by

Proof: Let A = (a, b) be the point on the unit circle above a, and let B = (c, d) be the point on the unit circle above c.

Let

be a partition of the arc where

The length of the inner approximation corresponding to this partition is

Since by the Lemma, there is a one to one correspondence between the partitions of the arc between (a, b) and (c, d) on the upper half of the unit circle and with the sets of the x coordinates of the points in the partition, to find the arc length, the least upper bound of the inner approximations for all of the partitions will be the same as the least upper bound of these sums over all sets off points on the x-axis

so we can take the arc length to be 

Factor out an x_i - x_i-1 from both terms under the radical.

Substitute for the ys

Rationalize the numerator

If we factor the difference of squares in the top, it will cancel with the factor in the bottom.

This is the definition of the following Riemann integral,

which will simplify to

Find common denominators

and the integral simplifies to

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