# 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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