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Evaluating integral for shell method example Video Lecture - Chemical Engineering
FAQs on Evaluating integral for shell method example
| 1. What is the shell method for evaluating integrals in chemical engineering? |  |
Ans. The shell method is a technique used in calculus to calculate the volume of a solid of revolution. In chemical engineering, it is often used to evaluate integrals when analyzing processes involving cylindrical or shell-shaped objects.
| 2. How does the shell method differ from other methods of evaluating integrals? | |
Ans. The shell method differs from other methods, such as the disk method, by using cylindrical shells instead of disks to approximate the volume of a solid. This makes it particularly useful when dealing with objects that have cylindrical or shell-like shapes.
| 3. Can you provide an example of how the shell method is applied in chemical engineering? | |
Ans. Sure! Let's say we have a cylindrical tank with a radius of 2 meters and a height of 5 meters. If we want to find the volume of the tank using the shell method, we would integrate the circumference of a shell (2πr) multiplied by its height (h) over the range of the tank's height (0 to 5 meters).
| 4. What are some common applications of the shell method in chemical engineering? | |
Ans. The shell method is commonly used in chemical engineering to analyze processes involving cylindrical objects such as tanks, pipes, and reactors. It can be used to calculate volumes, surface areas, and other important parameters in these systems.
| 5. Are there any limitations or considerations when using the shell method in chemical engineering? | |
Ans. While the shell method is a powerful tool in evaluating integrals for cylindrical or shell-shaped objects, it may not be suitable for more complex geometries. Additionally, it is important to carefully define the coordinate system and boundaries when applying the shell method to ensure accurate results.
Text Transcript from Video
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.
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.
About this Video
Apr 19, 2026 Last updated
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