Calc I- Evaluate an Integral using the Fundamental Th'm of Calculus - poyser

In this problem, we are going to evaluate this integral.
Now just a quick word of warning, and that is we have
to do a little FOIL action first.
That's going to be helpful to us, as well as this problem is
going to have heavy fractions in it and, at least, lots of
fractions in it and a little bit of algebra.
So hopefully, you're ready for that.
Now you can pause this video at any time, if you don't
understand something.
And as I'm going through this, I'm going to
go through it fast.
And I've got my calculator, got a calculator here off on
the side, so I'm going to be crunching
some numbers quickly.
So double-check my math here, OK?
But let's first start off by FOILing our integrand here.
We've got a binomial times a binomial.
So let's just work this out here first, OK?
What I have then is to the integral between 3 and 6.
If you take a moment and FOIL this and add up the like
terms, you should end up with negative x squared,
let's see, plus 9x--

all right, that's to the first power--
and then minus 18.
So if you take a quick second and FOIL all that, this is the
trinomial that you should end up with, all right?
So let's take and find the antiderivative here.
Maybe I should just do this in a block here.
Let's go and find the antiderivative of each of
these terms, OK?
Now I don't know how your teacher explains it to you,
but I usually, in class, start off saying, all right, you
want to find the antiderivative?
Then we start by adding 1 to this exponent.
This new exponent is a 3.
And that 3 moves to the bottom, it
also goes on the bottom.
So this first antiderivative is negative x to the
third all over 3.
OK, same thing here.
If there's a 1 as its exponent, and I'm just going
to add 1 to it, which now makes it a 2, and that 2 goes
to the bottom.
So this antiderivative is 9x squared all over 2.
And lastly, if you want to think of it as x and a 0
sitting there, you don't have to, but you can.
All right, you're going to add 1 to 0, which is 1.
And 1 moves to the bottom.
Well, that's not so bad.
So this one here has an antiderivative of
negative 18x, OK?
And all of this, we're going to evaluate between 3 and 6.
Now the first fundamental theorem of calculus says what
we're going to do is we're going to plug in 6 everywhere
I see an x in this answer.
And we're going to plug in a 3 everywhere we see
an x in this answer.
And we're going to subtract those two quantities, OK?
So here's the set up.
Are you ready?
All right, everywhere I see an x, I'm going to plug in a 6,
that is negative 6 to the third all over 3 plus 9, 6
squared all over 2, minus 18 times 6.
All right, everywhere I see an x, I'm plugging
in a 6 there, minus--
all right, it's always minus, that's what the first
fundamental theorem of calculus says--
minus everywhere I see an x, I'm going to plug in a 3 now.
So it becomes negative 3 cubed all over 3 plus 9, 3 squared
all over 2, and minus 18 times 3.
There.
I just barely squeezed it on the page.
OK, so again, this is everywhere I see an x, I'm
plugging in a 3, OK?
So if you work this all out--
and I'm going to do this pretty quickly here--
I believe the numerator for this guy turned into a
negative 216, because 6 to the third power is--
all right, real quickly here, 6 to the third
power, yep, is 216.
But don't forget the negatives sitting out in
front there, OK?
So this turns into negative 216 all over 3.
We're going to simplify it and clean this up
a little bit more.
Let's see, I've got 6 squared, which is 36.
And 1/2 of 36 is 18.
And 18 times 9 is going to give us--
I think this one here turns into a plus 162.
And again, this is all inside of a bracket here.
And 18 times 6, let's see, that's 108,
if I did that right.
Again, double-check my math, OK?
All right.
Let's see.
How about over here?
Let's clean this up.
3 to the third power is 27.
But 3 goes into 27 9 times.
But don't forget the negative.
So this first term turns into a negative 9 inside here.
3 squared is 9.
And 9 times 9 is 81.
81 divided by 2-- you know what?
I'm just going to leave it as a fraction, so plus 81 over 2.
And this last one here, 18 times 3, is 54, OK?
So let's go ahead and clean this up a little bit more.
I think I'm going to have to start another page here.
Scoot this up.
Now when doing this problem, do you see that--
let's see if you can still see this here--
do you see that negative 216 over 3?
Hey, 3 goes into that perfectly, right?
So this is going to turn into a negative 72
plus 162 minus 108.
Well, this whole thing here is simply going to turn into, I
believe, it's a negative 18, OK?
Again, double-check that math there.
And we'll just bring our minus sign on down.
Let's clean up our other bracket over here.

I can add all of these up together.
And I've got a negative 9 added to a--
maybe I could do the constants first, right?
A negative 9 added to a negative 54 is a negative 63.
So maybe you want to see it that way, negative 63
plus 81 over 2.
So the negative 63 came from negative 9 and negative 54.
To add these two up, I need a common
denominator of 2, right?
So let's do that.
Let's make this a 2 on the bottom, which means I need to
multiply the top by 2, which gives me a new numerator of
126, so negative 126 all over 2.
OK, and to clean this up now, look a negative 126 added to
an 81 is a negative 45, if I did that right.
So here's what I've got.
I've got negative 18 minus a negative 45/2.
Well, minus a negative is really the same thing as
plus-- here, we can get rid of this--
is really same thing as plus.
So I have negative 18 plus 45/2.
And if you work that out, you see, look, I need a common
denominator of 2.
Let's multiply the top by 2.
That becomes a 36.
So I've got negative 36 over 2 plus 45/2.
And my new numerator is just simply a 9, right?
Negative 36 added to 45 is 9.
So my final answer out of all of this is 9/2.
Wow.
If we were to actually graph this out, if we were to graph
out this function, which is what we have here in green,
this FOILed thing here, and we were to integrate it, we look
for the area under the curve between 3 6, we would get a
total area of 9/2, total area of 9/2 right there.
I hope that helps.