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# Part 2 of washer for non axis rotation

Video transcript

Where we left off
in the last video, we'd actually set up
our definite integral to figure out the
volume of this figure. So now we just have
to evaluate it. And really the
hardest part is going to be simplifying that
and that right over there. So let's get to it. So what is this thing squared? Well, looks like
we're going to have to do a little bit of
polynomial multiplication here. So I'll go into that
same color I had. So we're going to have 4 minus x
squared plus 2x times 4 minus-- actually, let me write that
in the order of the terms, or the degree of the terms. It's negative x squared
plus 2x plus 4-- I just switched the order of
these things-- times negative x squared plus 2x plus 4. So we're just going to
multiply these two things. I shouldn't even write
a multiplication symbol, it looks too much like an x. So 4 times 4 is 16. 4 times 2x is 8x. 4 times negative x squared
is negative 4x squared. 2x times 4 is 8x. 2x times 2x is 4x squared. And then 2x times
negative x squared is negative 2x to
the third power. And now we just have to multiply
negative x times all this. So negative-- or
negative x squared. Negative x squared times
4 is negative 4x squared. Negative x squared times 2x is
negative 2x to the third power. And then negative
x squared times negative x squared is
positive x to the fourth. So it's going to be
positive x to the fourth. And now we just have to
add up all of these terms. And we get, let's see,
we get x to the fourth-- add these two-- minus
4x to the third. And then this cancels
with this, but we still have that, so minus 4x squared. You add these two right
over here, you get plus 16x. And then you have plus 16. So that's this expanded out. And now if we want to-- 4
minus x times 4 minus x. So if we just have 4
minus x times 4 minus x, we could actually do
it this way as well. But that's just going
to be 4 times 4, which is 16, plus 4 times negative
x, which is negative 4x, negative x times 4,
another negative 4x, and then negative x times
negative x is equal to plus x squared. So if we were to
swap the order, we get x squared minus 8x plus 16. But we're going
to subtract this. So if you have the
negative sign out there, we're going to
subtract this business. So let's just do
it right over here since we already have it set up. So we have this minus this. We're going to
subtract this, or we could add the negative of it. So we'll put negative x
squared plus 8x minus 16. So I'm just going to add
the negative of this. And we get-- and I'll
do this in a new color-- let's see, we get x
to the fourth minus 4x to the third power minus
5x squared plus 24x, and then these cancel out. So that's what we are left with. And so that's the
inside of our integral. So we're going to take the
integral of this thing, just so I don't have to
keep rewriting this thing, from 0 until, if I
remember correctly, 3. Yup, from 0 to 3, of this, dx. And then we had a
pi out front here. So I'll just take that
out of the integral. So times pi. Well, now we just to
take the antiderivative. And this is going to be equal
to pi times antiderivative of x to the fourth is
x to the fifth over 5. Antiderivative of
4x to the third is actually x to the fourth. So this is going to be
minus x to the fourth-- you can verify that. Derivative is 4x, and
then you decrement the exponent, 4x to the third. So that works out. And then the antiderivative
of this is negative 5/3 x to the third. Just incremented the
exponent and divided by that. And then you have plus 24x
squared over 2, or 12x squared. And we're going
to evaluate that. I like to match colors
for my opening and closing parentheses. We're going to evaluate
that-- actually, let me do it as brackets. We're going to evaluate
that from 0 to 3. So this is going to be
equal to pi times-- let's evaluate all this business at 3. So we're going to get
3 to the fifth power. So let's see, 3 to the
third is equal to 27. 3 to the fourth is equal to 81. 3 to the fifth is going to be
equal to-- this times 3 is 243. So it's going to
be 243-- it's going to be some hairy math-- 243 over
5 minus-- well, 3 to the fourth is 81. Minus 81. Minus, let's see, we're going
to have 3 to the third times-- let's see, so it's going
to be minus 5 over 3 times 3 to the third power. So times 27. Well, 27 divided by 3 is just 9. 9 times 5 is 45. So let's just simplify that. So this is going
to be equal to 45. Did I do that right? Yeah. It's essentially going to
be like 3 squared times 5, because you're
dividing by 3 here. So it's going to have 45. And then finally,
3 squared is 9. 9 times 12 is 108. So plus 108. These problems that
involve hairy arithmetic are always the most
stressful for me, but I'll try not to
get too stressed. And then we're going to subtract
out this whole thing evaluated at 0, but lucky for us,
that's pretty simple. This evaluated is 0, 0, 0, 0, 0. So we're going to
subtract out 0, which simplifies
things a good bit. So now we just have to do some
hairy fraction arithmetic. So let's do it. So what I'm going to do first
is simplify all of this part. And then I'm going to
worry about putting it over a denominator of 5. So let's see what we have. We have negative 81 minus 45. So these two right over
here become negative 126. And then negative 126 plus 108. Well, that's just going to be
the same thing as negative 26 plus 8, which is going
to be negative 18. So this whole thing
simplifies to negative 18. Did I do that right? So this is negative 126. And then negative 126-- yes. It will be negative 18. So now we just have to
write negative 18 over 5. Negative 18 over 5 is the same
thing as negative-- let's see, 5 times 10 is 50, plus 40. So that's going to be
negative 90 over 5. So this whole thing
has simplified to-- is equal to pi
times 243 over 5 minus 90 over 5, which is
equal to pi-- guys, I deserve a drum roll now. This was some hairy mathematics. So 243 minus 90 is
going to be 153 over 5. Or we can write this
as 153 pi over 5. And after all that
math, you sometimes forget what we were
doing in the first place. We were figuring out
this right over here is the volume of
this figure that had this little inside
of it bored out.