What is the Relationship Between u-Substitution and Limits on an Integral?

[bobsmith76]

Homework Statement

I understand about everything except why the b and a values on the integral change from 0,3 to 9,36

 

[tiny-tim]

hi bobsmith76!

∫₀⁹ dx is an abbreviation for ∫_x=0^x=9 dx …

now convert x to u ! ​

 

[bobsmith76]

I know what it's an abbreviation for but how do you go from 0,3 to 9,36 by what rule is that legal?

 

[tiny-tim]

I know what it's an abbreviation for but how do you go from 0,3 to 9,36 by what rule is that legal?

uhh?

u = x³ + 9

so x = 0 -> u = 9,

so x = 3 -> u = 36 ​

 

[bobsmith76]

ok, thanks

 

[bobsmith76]

Does this rule have a name so that I can look it up in my book? I don't see why u-substitution should be related to the a and b values on an integral.

 

[Dick]

Does this rule have a name so that I can look it up in my book? I don't see why u-substitution should be related to the a and b values on an integral.

I don't think it has any particular name. But if you express your final integral as a function of u, then the limits have to change to the limits for u. If you don't like that, then change the u back into x^3+9 at the end, so you've got 6π(x^3+9)^(1/2). Now use the original limits. It's exactly the same thing.

 

[tiny-tim]

(just got up :zzz: …)

Does this rule have a name so that I can look it up in my book? I don't see why u-substitution should be related to the a and b values on an integral.

yes … and the name is u-substitution!

if you substitute u for x, you must do so wherever x occurs,

including in the the limits!

u-substitution does exactly what it says on the tin! ​