What is the average volume of a cube inside the unit cube if all the faces are parallel to the coordinate planes? Here are the results of an experiment with trials repeated 10 times.
0.0283584 | 0.0285983 | 0.0286288 | 0.0284869 | 0.0283728 |
0.0285560 | 0.0284275 | 0.0285388 | 0.0283703 | 0.0289527 |
Let denote the edge of a cube. A cube of this size can be moved so that the vertex closest to the point
can be placed anywhere inside the small cube, which has edge
.
The expected volume is given by the integral