The Spherical Crust Problem

Prove: If you have a spherical loaf of bread, and put it in a deli bread slicer to ensure that each slice has the same thickness, each slice will have the same area of crust.

The diagram above shows a quarter of a circle whose equation is:

                                     

We wish to find the surface area of the crust obtained by rotating the shaded region around the x-axis.  This would correspond to the area of crust in one slice of thickness k.  This can be done by first looking at a small piece of the arc length at the top of the shaded region.  This is given by:

To turn this bit of arc into a bit of crust area, we need to spin it around the x-axis.  This is done by simply multiplying by 2py. 

Therefore, DA = 2paDx.  To get the total crust from b to b+k exactly, we need to sum over all the DA's in this region, allowing Dx to go to zero.  We express this as the integral:

This means that the area of the crust is independent of b (where you start cutting), and only on k, the thickness.  Thus all slices of thickness k will yield the same crust.