First of all, the things that you actually integrate  are  densities, which  are the differential geometric counterparts  of  measures.   No orientation is needed.   

 A degree $n$ form on an $n$-dimensional manifold is almost  a density, but not quite. We need an orientation to  associate to the top degree form a  density.  This  is what you ultimately integrate when you integrate  a  form.  For more details see  page 105 of [these notes][1].


  [1]: http://www.nd.edu/~lnicolae/Lectures.pdf