Question

Why substitution when u' changes sign doesn't work on definite integrals?

From Unit3 Part A we learn how to compute definite intergals using substitution and changing the limits of the integral. We learn this doesn't work if u' changes sign, but I don't really get why is that?

Related course note: http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-3-the-definite-integral-and-its-applications/part-a-definition-of-the-definite-integral-and-first-fundamental-theorem/session-49-applications-of-the-fundamental-theorem-of-calculus/MIT18_01SCF10_Ses49c.pdf

Answer 1

Here's one way to think about it.

Answer 2

Thank you phi! I understood thanks to the explanation stating that if dx changes sign, then obviously the sum represented by the integral turns wrong! That's what I was missing, the sign of dx matter obviously, got it now, that's great. :)

BTW the document is very well explained, did you find out somewhere or wrote it yourself?

Answer 3

Also, would you recommand to convert back with the original variable when possible, instead of changing the limit of the integral? To avoid errors.

Answer 4

I wrote it up, as an exercise in explaining what is going on.
Generally, people change the limits to match the new variable.

Answer 5

Ok, thanks for your time and answers!

Answer 6

yw