I received a suggestion that I do actual old AP exam problems,
and I looked on the internet and lo and behold, on the
college board site, if you go to collegeboard.com, you can
actually get-- I couldn't find the actual multiple choice
questions, but you can find the free response questions, and so
this question is actually the first free response question
that they have on the calculus BC that was administered
just recently in 2008.
So let's do this problem.
And frankly, if you understand how to do all of the free
response questions, you probably will do fairly well on
the multiple choice, because the free response tend to be a
little bit more challenging, especially the last parts
of the free response.
Well anyway, let's do this one.
So I'll just read it out, because I don't want to write
it out all here, but this is the actual diagram.
I actually copied and pasted this from the PDF that they
provide on collegeboard.com.
So it says, let r-- this is r-- be the region bounded by the
graphs of y equals sine pi of x.
So let me write that down.
So this top graph is y is equal to sine pi x.
and then the bottom graph is y is equal to x cubed minus 4x.
And how did I know that this was the bottom one?
Well I knew that this one was sine of pi x, right?
Because sine looks like this.
It doesn't look like that, right?
When you go sine of pi is 0, sine of 0 is
0, sine of 2pi is 0.
So we do this as sine of pi x.
Well anyway, they want-- so this is the region between
these two functions and part A of this-- and this is kind of
the softball question, just to make sure that you know how to
do definite integrals-- and it says, find the area of r.
So how do we do that?
I think you know that we're going to do a little definite
integration, so let's do that.
So then we're going to take the definite integral, so let's
just say the area is equal to-- I don't know if that's-- I hope
I'm writing big enough for you-- the area is going to be
equal to the definite integral from.
So what are the x values?
We're going to be going from x is equal to 0
to x is equal to 2.
And what's this?
At any given point value of x, what is kind of going to be the
high-- when we're taking the area, we're taking a bunch of
rectangles that are of dx width, right?
So that's-- that's not dark enough, I don't think that
you can see that-- so that's one of my rectangles.
Whoops.
Let's say that's one of my rectangles right here that
I'm going to be summing up.
Its width is dx.
What's its height?
Its height is going to be this top function minus
this bottom function.
So, essentially, we're going to take the sum of all of these
rectangles, so its height is going to be-- let me switch
colors arbitrarily-- the height is going to be the top function
minus the bottom function.
So sine of pi x-- parentheses here-- minus the
bottom function.
So minus x cubed plus 4x.
Since I'm subtracting, I switched both of these signs.
And all of that times the width of each of these little
rectangles-- which is infinitely small-- dx.
And we're going to sum them all up from x is equal
to 0 to x is equal to 2.
This should be fairly straightforward for you.
So how do we evaluate this?
Well, we essentially take the antiderivative of this and
then evaluate that at 2 and then evaluate at 0.
What's the antiderivative of sine of pi x?
Well, what functions derivative is sine of x.
Cosine of x-- let's see.
If I were to take the derivative of cosine-- let's
say I took the derivative of cosine pi x.
This should be reasonably familiar to you.
Cosine of pi x, if I were to take the derivative
of it, what do I get?
That equals pi.
You take the derivative of the inside, right?
By the chain rule.
So it's pi times the derivative of the whole thing.
The derivative of cosine of x is minus sine of x, so the
derivative to this is going to be times minus sine of pi x, or
you could say that equals minus pi sine of pi x.
So the derivative of cosine of pi x is almost this, it just
has that minus pi there, right?
So let's see if we can rewrite this so it looks just like the
derivative of cosine pi x.
And I'll switch to magenta.
I want to make sure I have enough space to do
this entire problem.
So let's write a minus 1 over pi times a minus pi.
All I did, when you evaluate this, this equals 1, so I can
do this times sine pi x, and then that's minus x to the
third plus 4x, and then all of that times the width dx.
Well now we have it.
We know that the antiderivative of this is cosine pi x, right?
And this is just a constant term.
So what's the antiderivative of this whole thing?
And I'll arbitrarily switch colors again.
The antiderivative is cosine pi x.
So we have minus 1 over pi cosine pi x-- remember, I could
just carry this over, this is just a constant term-- this
antiderivative is this right here.
And then these are a little bit more straightforward.
So minus the antiderivative of x to the third is x to the
fourth over 4 plus the antiderivative of this is 4x
squared over 2, or you could just view that as 2x squared,
and then we're going to evaluate that at 2 and at
0, and let's do that.
So this is equal to cosine of 2pi, and we'll have a minus
sign out here, so minus cosine of 2pi over pi, minus-- what's
2 to the fourth power?
Let's see.
2 to the third is 8, 2 the fourth is 16, 16 over 4 is 4,
so it's minus 4, 2 squared is 4 times 2 is 8, so plus 8, so
that's the antiderivative evaluated at 2, and now let's
subtract it evaluated at 0.
So this will be minus cosine of 0 over pi-- all right, that's
that evaluated at 0-- minus 0, plus 0.
So these terms don't contribute anything when
you evaluate them at 0.
And so what do we get?
What's cosine of 2pi?
Cosine of 2pi is the same thing as cosine
of 0, and it equals 1.
What is the x value of the unit circle at 2pi, or at 0?
It's equal to 1.
So this equals minus 1 over pi minus 4 plus 8, and so this
minus minus, those both become pluses, cosine of 0 is also 1,
so plus 1 over pi, and so this minus 1 over pi and this plus 1
over pi will cancel out, and all we're left with is minus 4
plus 8 and that is equal to 4.
So that is part one, part A of number one, on the 2008 DC
free response questions.
It actually took me a whole video just to do that part.
In the next video, I'll do part B, and we'll just keep doing
this, and I'll try to do a couple of these every day.
See you soon.