Kenneth S. answered • 12/05/16
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As x varies from 0 to 1, (1-x) varies from 1 to 0, so if one goes back to the definition of the definite integral as the limit of a Riemann summation, the identical succession of infinitesimally narrow rectangles (length of base is dx) will express the same 'altitudes' (values of y), but in reverse order. Therefore these limits will be identical.
Kenneth S.
Actually, I think that what I wrote is acceptable proof. Proofs just have to make sense, and have sufficient reasoning in them. To do this more laboriously using purely mathematical notation would be hideously error-prone, on this platform. (By this I mean it would be slow typing and very error prone, having to switch to subscripts and back, etc.)
Now that I think of it, you can do a u-substitution and reverse the upper & lower limits of the definite integral; let u = 1-x so du = -dx and when x=1,u=0 and vice versa...
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12/07/16
Harris C.
12/06/16