[music] What do we get if we integrate an odd function?
For instance, what happens when we integrate the function sin x from minus 1
to 1, right, sin x is an odd function. Consequently, it looks like the answer is
0. Remember the integral is, is morally
calculating an assigned area. So it's going to be calculating this area
here but with a negative sign. And it's going to be calculating this area
here as actual area. Right?
With a positive sign. And these two areas are equal, but the
integral is going to count this area with negative sign and this area with a
positive sign. So the integral should cancel and the
answer should be 0. But if you don't believe the geometry, we
can work it out algebraically. So algebraically, I'm trying to integrate
from minus 1 to 1 of sine x dx. And the first thing to think is that well
if I'm integrating from minus 1 to 1, can we write that as an integral of minus 1 to
0 of sin x dx, and add to that the integral from 0 to 1 of sin x dx right.
Right, if I want to integrate all the way from minus 1 to 1, it's good enough if I
just integrate half way and then integrate the rest of the way, and there is two
separate intervals. Now, how do I integrate from minus 1 to 0
sin x dx? Now if we think a little bit about what
this really means, that would be the same as integrating from 0 to 1 of sin minus x,
alright? If I want to add up numbers from minus 1
to 0, it'll be good enough if I pick the partition from 0 to 1 and then evaluate
the sin at negative, those, those value. And you have to worry a little bit about
how this dx changes but we are going to be looking at that in the future as well.
Okay, and I'm going to add to this the interval from 0 to 1 of sine x dx that
I've got right there. And what's sine of negative x.
That's negative sine of x, and here I've just got sine of x dx again.
But now I've got the interval from 0 to 1 of negative sine dx plus the interval from
0 to 1 of sine dx. Well I can pull this minus sign out of the
integral. If I'm integrating just a constant time
something it's the same as that constant times the integral.
And now I've got negative something plus the same thing so that is equal to 0.
Now in the future we're going to see more how to justify those steps that I took
algebraically, right? Why are those steps actually valid.
But in the meantime, the upshot here, the takeaway message is just that symmetry can
be exploited to calculate some integrals, right.
You can evaluate the integral of an odd function, as long as you're integrating
symmetrically across 0, you can evaluate that integral to be equal to 0 by
exploiting symmetry.