Re-ordering a Triple Integral

(This is example 5.7 in the APSC 172 workbook) We're given a triple integral and asked to change the order of integration. This is an excellent type of problem to test our understanding of multiple integrals, and I try to go through it carefully. Note: the problem is solved as of the 09:45 mark; after that, we just check our answer, and I draw our attention to the possibility of making subtle sign mistakes when working with fractional powers.