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 of 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.