# Simple Question about integral definition

1. Jul 30, 2013

### Andrax

1. The problem statement, all variables and given/known data
so this is my first time learning about integrals , from spivak' calculus
Actual quote : the integral $$\int_{a}^{b} f(x) \, \mathrm{d}x$$was defined only for a<b we now add the definition
$$\int_{a}^{b} f(x) \, \mathrm{d}x$$=-$$\int_{b}^{a} f(x) \, \mathrm{d}x$$ if a>b "
isn't he contradicting himself here to write$$\int_{a}^{b} f(x) \, \mathrm{d}x$$ a<b is required right?so you can't just write $$\int_{a}^{b} f(x) \, \mathrm{d}x$$ when yo usay "if a >b"
i tried doing problem 7 which involves the function x^3
we have $$\int_{-1}^{1} x^3 \, \mathrm{d}x$$=$$\int_{-1}^{0} f(x) \, \mathrm{d}x$$ + $$\int_{0}^{1} f(x) \, \mathrm{d}x$$(so far everything is normal) =applying spivak's definition -$$\int_{-1}^{0} f(x) \, \mathrm{d}x$$ +$$\int_{0}^{1} f(x) \, \mathrm{d}x$$ why in the answer books he says this equals 0 ? this dosen't make sense at all since [0;-1] is not an interval?$$\int_{-1}^{0} f(x) \, \mathrm{d}x$/[itex] requires that 0<-1 .. Please help i am VERY confused. 2. Relevant equations mentioned above 3. The attempt at a solution mentioned above Last edited: Jul 30, 2013 2. Jul 30, 2013 ### harikasri It is an odd function. 3. Jul 30, 2013 ### Ray Vickson What's the problem? -1 < 0 is certainly true! Why would you think otherwise? 4. Jul 30, 2013 ### Andrax well i miss typed that, i mean 0<-1 laughs , well i think none is getting me here? can someone prove that [itex]$\int_{-1}^{1} x^3 \, \mathrm{d}x$$ = 0 ? by splitting the intervals to -1 , 0 and 0 1 , that would help clear these things

5. Jul 30, 2013

### LCKurtz

If $f$ is an odd function so $f(-x) = -f(x)$ then for $a>0$,$$\int_{-a}^af(x)\, dx =\int_{-a}^0f(x)\, dx + \int_0^a f(x)\, dx$$Let $x=-u$ in the first integral making it$$\int_{a}^0f(-u)\, (-1)du = \int_{a}^0 -f(u)\, (-1)du = \int_a^0f(u)\, du =-\int_0^af(u)\, du$$so this integral cancels the second integral, giving $0$.

6. Jul 30, 2013

### Andrax

thanks everything is clear now and a little bit offtopic , wow Integral is hard compared to derivatives and limits i might switch to another book spivak became suddenly very hard to me

7. Jul 30, 2013

### Staff: Mentor

Some LaTeX tips.
You're putting in way more symbols than you actually need - extra brackets and slashes.
Use a single pair of LaTeX delimiters for an entire equation, rather than breaking it up into multiple LaTeX expressions.

Instead of writing this: $$\int_{a}^{b} f(x) \, \mathrm{d}x$$, you can write it much more simply this way: $\int_a^b f(x)~dx$
Code (Text):
$\int_a^b f(x)~dx$
Or instead of this: $$\int_{a}^{b} f(x) \, \mathrm{d}x$$=-$$\int_{b}^{a} f(x) \, \mathrm{d}x$$
You can write this:
$\int_a^b f(x)~dx = -\int_b^a f(x)~dx$
Code (Text):
$\int_a^b f(x)~dx = -\int_b^a f(x)~dx$