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[music] What do we get if we integrate an
odd function?

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For instance, what happens when we
integrate the function sin x from minus 1

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to 1, right, sin x is an odd function.
Consequently, it looks like the answer is

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0.
Remember the integral is, is morally

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calculating an assigned area.
So it's going to be calculating this area

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here but with a negative sign.
And it's going to be calculating this area

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here as actual area.
Right?

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With a positive sign.
And these two areas are equal, but the

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integral is going to count this area with
negative sign and this area with a

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positive sign.
So the integral should cancel and the

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answer should be 0.
But if you don't believe the geometry, we

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can work it out algebraically.
So algebraically, I'm trying to integrate

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from minus 1 to 1 of sine x dx.
And the first thing to think is that well

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if I'm integrating from minus 1 to 1, can
we write that as an integral of minus 1 to

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0 of sin x dx, and add to that the
integral from 0 to 1 of sin x dx right.

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Right, if I want to integrate all the way
from minus 1 to 1, it's good enough if I

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just integrate half way and then integrate
the rest of the way, and there is two

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separate intervals.
Now, how do I integrate from minus 1 to 0

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sin x dx?
Now if we think a little bit about what

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this really means, that would be the same
as integrating from 0 to 1 of sin minus x,

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alright?
If I want to add up numbers from minus 1

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to 0, it'll be good enough if I pick the
partition from 0 to 1 and then evaluate

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the sin at negative, those, those value.
And you have to worry a little bit about

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how this dx changes but we are going to be
looking at that in the future as well.

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Okay, and I'm going to add to this the
interval from 0 to 1 of sine x dx that

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I've got right there.
And what's sine of negative x.

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That's negative sine of x, and here I've
just got sine of x dx again.

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But now I've got the interval from 0 to 1
of negative sine dx plus the interval from

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0 to 1 of sine dx.
Well I can pull this minus sign out of the

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integral.
If I'm integrating just a constant time

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something it's the same as that constant
times the integral.

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And now I've got negative something plus
the same thing so that is equal to 0.

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Now in the future we're going to see more
how to justify those steps that I took

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algebraically, right?
Why are those steps actually valid.

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But in the meantime, the upshot here, the
takeaway message is just that symmetry can

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be exploited to calculate some integrals,
right.

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You can evaluate the integral of an odd
function, as long as you're integrating

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symmetrically across 0, you can evaluate
that integral to be equal to 0 by

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exploiting symmetry.