An Exclusive Integral

Calculus Level 5

The bitwise exclusive or operator "\(\wedge\)" is well-defined for all non-negative integers. We can easily extend it to the non-negative reals. Simply take two non-negative reals, \(x\) and \(y\), and write each one out in binary. Line the digits up, and let \(z = x \wedge y\) like so:

\(x = 101.001101010100110101010011010101...\) \(y = 111.000011000011001111111111011111...\) \(z = 010.001110010111111010101100001010...\)

It may be the case that \(x\) or \(y\) have two possible representations in binary, one ending in infinite \(0\)'s, the other in infinite \(1\)'s. For uniqueness and therefore determinism, we choose the representation with infinite \(0\)'s.

Let \(C\) be the interior of a circle in \(\mathbb{R}^2\), with radius \(\frac{1}{2}\), and center at \((\frac{1}{2}, \frac{1}{2})\). Let \(I\) as:

\(I = \iint\limits_{C} x \wedge y\:dx\:dy\)

What is \(2^{20} I\), rounded to the nearest integer?


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