Hovedinnhold

## Volume: shell method (optional)

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# Regne ut integraler ved bruk av skallmetoden - eksempel

## Video transkripsjon

Where we left off
in the last video, we had set up a
definite integral using the shell method for this
strange solid of revolution. So now, let's just
evaluate the integral. And like we've seen many times
in these type of problems, we really just have to do
some polynomial multiplication right over here. So x minus 3 squared, well,
that's pretty straightforward. That's going to be x
squared minus 6x plus 9. And we're going to multiply
that times x minus 1. So let's do that first. So multiply that
times x minus 1. And so negative 1
times 9 is negative 9. Negative 1 times negative 6
is positive 6, positive 6x. Negative 1 times x squared
is negative x squared. Now x times 9 is 9x. x times negative 6x is
negative 6x squared. And then x times x
squared is x to the third. And so we get x
to the third minus 7x squared plus 15x minus 9. So we just multiplied x minus
3 squared times x minus 1, and then we have to
multiply that times x. So we could essentially
raise the degree of each of these things. At least it's easier now
take the anti-derivative. It's equal to 2 pi times the
definite integral from 1 to 3 of this stuff times x. So it's going to be
x to the fourth minus 7x to the third power
plus 15x squared minus 9x. And then, of course, dx. And I'll make the dx in
that same nice blue color. Now let's just take
the anti-derivative. So this is going to be equal to
2 pi times the anti-derivative of all of this business. We're going to evaluate
it at 3 and subtract it when it's evaluated at 1. So the anti-derivative
of x to the fourth is x to the fifth over 5. The anti-derivative
of x to the third is x to the fourth
over 4, and we're going to multiply
that times negative 7. So it's negative 7x
to the fourth over 4. And then the anti-derivative
of 15 x squared, that's going to be 15 times
x to the third over 3. 15 divided by 3 is 5, so
it's plus 5x to the third. And then finally,
the anti-derivative of negative 9x, that's going to
be negative 9x squared over 2. And you can verify. If you take the
derivative of this, you get this business
right over here. And so this is going
to be equal to 2 pi. And so let's evaluate all
of this business at 3. So when you evaluate
it at 3, you have 3 to the
fifth power over 5. And I believe 3 to the fifth
is 243, but I'll verify. 3 to the third is 27, 3 to
the fourth is equal to 81, 3 to the fifth is 243. So this is going to give us
some hairy math to deal with. So it's going to be 243 over 5. 3 to the fourth
power, that's 81. But then we have to
multiply 81 times 7. So we're going to get 567. Is that right? 81 times 7. 7 times 1 is 7, 7 times 8 is 56,
so we're going to get minus 567 over 4. This is going to be really
painful to do the arithmetic part. But we'll power through it. And then we have 5x
to the third is 27. 27 times 5 is what? 135? I don't want to
make any mistakes. 27 times 5. 7 times 5 is 35,
2 times 5 is 10. Yep. 135. So plus 135. And then finally, we
have minus 9x squared. So x squared is 9 times 9 is 81. So minus 81 over 2. So that's all of this
business evaluated at 3. And from that, we're
going to subtract it when it's evaluated at 1. Let's do this. So we get 1/5 minus 7/4
plus 5, and then minus 9/2. And what we are left with is
just a really hairy fractions problem. So I will just hope that I
don't make a careless mistake at this point. So let's try to do this. This is going to be
equal to 2 pi times, and if we wanted to find
a common multiple here, it looks like it
would have to be 20. Least common multiple
of 5 and 4 and 2 is 20. So this is going to give us
243 over 5 is the same thing. 243 times 4 is going to give
us 3 times 4 is 12, 4 times 4 is 16, plus 1 is 17. 2 times 4 is 8, plus 1 is 9. So we have 972 over 20. And then we have to
multiply 567 times 5. So you can see the arithmetic
is the most painful part here. 7 times 5 is 35, 6 times 5 is
30, plus 3 is 33, 5 times 5 is 25, plus 3 is 28. So we have 2,835 over
4, and then 135 over 20. Well, 135 times 2 is going to be
270, and then times another 10 is 2,700. So plus 2,700 over 20. Did I do that right? Yeah, that's right. And then finally, 81 over 2. That's going to be the
same thing as negative 810. Let me do that same color. Negative 810 over 20. Numerator and denominator
both multiplied by 10. And then let's see. Negative 1/5, that's the
same thing as negative 4/20. It's going to be positive
7/4 is the same thing as positive 35/20. And it's going to
be a negative 5 is the same thing as
negative 100 over 20. And then finally, it's
going to be a positive. I don't want make
that careless mistake. I want to make sure I
get the signs right. After I distribute
this negative, it's going to be a
positive 9/2, which is the same thing
as plus 90 over 20. Did I do all the signs right? Negative 1/5, positive for
this one, so positive 7/4, negative 5, and then
positive 90 over 20. And so now, I just have to
do some hard core addition. So let's do it. So first, I'll take
all of the positives and then I'll subtract out the
negatives, just to simplify it, so I have to minimize
the number of times. Well, I'll add all the
positives together, and then I'll add all
the negatives together. And that ought to make it
one subtraction problem. So 972 plus 2,700
plus 35 plus 90. So let me just write it down. So this is 2,700 plus 972. I should probably take out
a calculator at this point, but I'll just do it by
hand, since I've already done so much of it by hand. Plus 972 plus 90 plus 35. So we get a 7. 7 plus 9 is 16, plus 3 is 19. Did I do that right? Yeah, 16 plus 3 is 19, and
this is 17, and this is a 3. So we have 3,797 when we add
in all the positive numerators. And then all the negative
numerators, let's see. I'm going to add them
together to see how negative. So 2,835, 810, 4, and 100. So if I add 2,835, 810, let
me see, 100, and 4, this is how much negative I
have to subtract from that. 5 plus 4 is 9, 3 plus 1 is
4, 8 plus 8 plus 1 is 17, and then you have a 3. So we're going to
subtract 3,749 from 3,797. And so this actually
works out quite well. That gets us to 48. Let me make sure I haven't
made a careless mistake. 2,835, 810, 100, and 4. Those are all the
things I'm subtracting. 2,700, 972, 90, and 35 are
all the things I added. Yep. 3,797 minus 3,749 is
going to be equal to 48. So this whole expression--
we deserve a drum roll now-- is going to be equal to
2 pi times 48 over 20. And both 48 and 20
are divisible by 4. So you get 12 over 5. My brain is turning
into mush now. I'm becoming paranoid that I've
been making careless mistakes. We're almost there. So it becomes 12 over 5. And so our final answer,
12 times 2 is 24. So it becomes 24 pi over 5. And we are done.