PROFESSOR: Hi.
Welcome back to recitation.
In lecture you've been doing definite integration for the
last couple of lectures.
And you've just started with the
fundamental theorem of calculus.
So, that gives us this whole powerful tool to compute a
bunch of integrals.
So, with that in mind I've got a couple questions for you.
So compute the definite integrals.
The integral from pi over 6 to pi over 3 of tan x dx.
And the integral of minus pi over 3 to pi
over 3 of tan x dx.
So why don't you pause the video, take a couple minutes
to work on those, come back and we'll
work on them together.
All right.
Welcome back.
So, as we were saying we have a couple of
definite integrals here.
So one tool for definite integrals is Riemann sums. So,
in principle, we could write down a Riemann sum for, say,
this first integral here.
And then try and compute it by trying to compute the sum and
taking a limit of it.
But I think it's pretty clear that, for a function like tan
x, that's going to be really, really hard to do.
So we want to not use Riemann sums here.
And the way we can not use Riemann sums is using this
great tool that we have, which is the
fundamental theorem of calculus.
So the fundamental theorem of calculus says, when you have a
definite integral, instead of computing Riemann sums, what
you can do is compute an antiderivative and use the
antiderivative to find out what the integral is.
Just compare two values with the antiderivative and that'll
give you the value of the definite integral.
So, in particular, let's do this first integral first. So
we have the integral from pi over 6 to pi
over 3 of tan x dx.
Well, so, we know what the antiderivative of tan x is.
We did that in an earlier recitation.
So that's ln of cosine x.
Or ln of the absolute value of cosine x.
So, the fundamental theorem of calculus says that the value
of this definite integral, in order to compute it, we just
take the difference of that antiderivative at pi over 3
and at pi over 6.
So, by the fundamental theorem of calculus this is equal to
ln of the absolute value of cosine x for x between pi over
6 and pi over 3.
So, OK, so this, you know, this is this notation that
Professor Jerison introduced.
So what does this actually mean?
It's just a shorthand for ln of the absolute value of
cosine of pi over 3 minus ln of the absolute value of
cosine of pi over 6.
Now, we could leave our answer like this but we can also
manipulate it and put it in a little nicer form.
So, cosine of pi over 3 is--
oh, I'm sorry I've made a mistake.
So it's not ln of absolute value cosine x.
It's minus ln of absolute value of cosine x.
Sorry about that.
So this is minus ln cosine pi over 3 plus--
OK, so minus and minus there is plus.
All right.
Sorry about that.
OK, so, good.
All right.
And we can pick up where I was in the middle of my sentence
and OK, so we can just evaluate this by just
evaluating.
So cosine of pi over 3, that's going to be 1/2.
So this is minus ln and absolute value of
1/2 is just a 1/2.
Plus cosine of pi over 6 is square root of 3 over 2.
So plus ln square root of 3 over 2.
And now you can use your logarithm rules to combine
those into a single expression.
So this is equal to ln of square root of 3.
Or you could even use one more logarithm rule if you wanted.
So this is equal to 1/2 ln of 3.
So all of those are, you know, equivalent expressions for the
same number, which is the value of
this definite integral.
OK, so, for the second integral now, we can do
exactly the same thing if we wanted to.
And it'll be a very similar process.
You'll write down this antiderivative here, you'll
take the difference in values.
We can also do something a little bit
clever for this one.
Which is, definite integrals have some geometric
interpretation, right?
And if you think about the function tan x--
so we could put up a little graph here--
so in between minus pi over 3 and pi over 3, tan x maybe
looks something like this.
Yeah, you know, give or take.
OK, so what's important about this?
Well tan x is an odd function.
And this interval from--
this is minus pi over 3 to pi over 3, and this is the curve
y equals tan x.
So this is an odd function.
So the integral in question is this positive area minus this
negative area.
But since it's an odd function, it's symmetric.
So those two cancel out perfectly.
So for the second one, you could do the same
process that we did.
But you can also use a little bit of geometric reasoning to
realize that this second one is just equal to 0.
In either case, doing either of these integral with Riemann
sums would be really atrocious.
So, in both cases, what we're really happy about is that we
have these other tools that allow us to evaluate definite
integrals without going back to the Riemann definition of
the integral.
So, I'm going to stop there.