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Really need help with Calculus problem?! 10 points?
I have a proof that I could use help with...
Use substitution to prove that the integral from -a to a of f(x)dx = 0 if f is an odd function.
It is pretty straight forward. The lower limit of integration is -a and the upper is a, just in case I said it wrong earlier haha. It would help if someone could at least tell me what an odd function is... I'm lost.
1 Answer
- PaulLv 48 years agoFavorite Answer
An odd function is one such that for all x in its domain, f(-x) = - f(x).
With regard to evaluating the integral on the interval S0 = (-a, a), split it into the two intervals S1 = (-a,0) and S2 = (0,a) *
I will use Int_S1 [f(x)dx] to denote the definite integral from -a to a, etc.
Now, Int_S0 [f(x)dx]
= Int_S1 [f(x)dx] + Int_S2 [f(x)dx]
= - Int_S1 [- f(x)dx] + Int_S2 [f(x)dx]
= - Int_S1 [f(-x)dx] + Int_S2 [f(x)dx]
= - Int_S2 [f(x)dx] ** + Int_S2 [f(x)dx]
= 0
* I am just using this notation because it is easier to type into the computer and it can be easily identified. .
** The integral of f(-x) on the interval (-a,0) is equal to the integral of f(x) from "a" to "0". Also the integral of f(x) from "a" to "0" is the opposite of the integral of f(x) evaluated from "0" to "a".
