Question about Substitution in Integration - Change in integral limits (1 Viewer)

[blackops23]

Hi just a quick question, to illustrate my problem I'm going to use the example:

Integration of x(4-x)^0.5............. domain: -5(<)x(<)3

Now using substitution of u=4-x, lets proceed

So when x=3, u=1
when x=-5, u=9

Now when you are changing the limits from "respect to x" to "in respect to u", do you put the 1 on the top of the integral sign and 9 on the bottom,

OR

Do you put the 9 on the top of integral sign, and the 1 on the bottom, as 9 is larger than 1.??

So which one do you do?

Thanks, appreciate the help

 

[slyhunter]

1 on top, 9 on bottom

 

[Omnipotence]

^ Yep. Alternatively you could put a negative sign in front of the integrand and reverse the paramater with 9 on top, 1 on bottom.

 

[funnytomato]

also don't forget du=-dx

 

[blackops23]

^ Yep. Alternatively you could put a negative sign in front of the integrand and reverse the paramater with 9 on top, 1 on bottom.

how does this work?

 

[slyhunter]

Integral limits flip if the integral becomes negative and vice versa.

 

[Omnipotence]

Its a rule?

 

[Trebla]

It is simple to prove. By the fundamental theorem of calculus


Also, the upper limit does not necessarily have to be greater than the lower limit. In a substitution, your limit transformation should always be corresponding to the original limits (i.e. upper limit in x --> upper limit in u & lower limit in x --> lower limit in u)