What fraction of a cube is closer to center than any surface

Calculus Level pending

The same method was used as was used in "What fraction of a regular tetrahedron is closer to center than any surface".

This problem's question is its title. Since the volumes of both the cube and the interior closer space scale with the cube of the edge the fraction remains the same regardless of the chosen edge length. The answer is in parts per billion rounded to an integer, i.e., Round[1000000000 value].

In the solution the integral and a closed form are given.


The answer is 86841005.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

∫ 1 4 ( 1 − 3 ) 0 ∫ 1 2 \ ( 1 − 2 − 4 x 2 ) x ∫ x 2 + y 2 − 1 4 y 1 d z d y d x \int _{\frac{1}{4} \left(1-\sqrt{3}\right)}^0\int _{\frac{1}{2} \ \left(1-\sqrt{2-4 x^2}\right)}^x\int _{x^2+y^2-\frac{1}{4}}^y1dzdydx

1 384 ( − 9 3 + 2 π + 10 ) 1 48 → 1 8 ( − 9 3 + 2 π + 10 ) ≈ 86841005 \frac{\frac{1}{384} \left(-9 \sqrt{3}+2 \pi +10\right)}{\frac{1}{48}}\to \frac{1}{8} \left(-9 \sqrt{3}+2 \pi +10\right) \approx 86841005

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