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HERBERT GROSS: Hi.
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In our last few lectures we were
trying to establish the
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identity of integral calculus
and differential calculus in
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00:00:39,810 --> 00:00:43,730
their own right, independently
of one another, and then by
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the fundamental theorems of
integral calculus to show the
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amazing relationship between
these two subjects.
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00:00:50,300 --> 00:00:54,680
Now what we would like to do
today is to emphasize this
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00:00:54,680 --> 00:00:58,330
topic in terms of an application
unlike what we've
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00:00:58,330 --> 00:00:59,600
been doing before.
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00:00:59,600 --> 00:01:04,220
In particular, what we will do
today is discuss the question
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of finding volumes.
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And in doing this, several
interesting things should
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happen, not the least of which
is that we will rederive
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00:01:12,310 --> 00:01:15,140
certain results that we've been
taking for granted about
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00:01:15,140 --> 00:01:17,180
solid regions--
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00:01:17,180 --> 00:01:19,060
volumes of regions--
for quite awhile.
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00:01:19,060 --> 00:01:22,380
And also, it will give us an
excellent chance to understand
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00:01:22,380 --> 00:01:25,640
what we really mean by a
mathematical structure.
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00:01:25,640 --> 00:01:28,330
Well, at any rate, to emphasize
the structure part,
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00:01:28,330 --> 00:01:32,690
I've called today's lesson,
'3-dimensional Area'.
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00:01:32,690 --> 00:01:35,400
See, instead of calling
it volume, I call it
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00:01:35,400 --> 00:01:38,580
3-dimensional area, and the
reason for this is I'd like to
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00:01:38,580 --> 00:01:41,860
show you how one can study
volumes in a completely
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00:01:41,860 --> 00:01:44,800
analogous way to how
we studied areas.
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00:01:44,800 --> 00:01:47,480
Remember, we had three basic
assumptions for area.
35
00:01:47,480 --> 00:01:52,960
I'm now going to assume three
similar assumptions for volume
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00:01:52,960 --> 00:01:55,930
except where I have to amend
them by necessity.
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00:01:55,930 --> 00:01:58,630
And the only place this
amendment has to take place is
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00:01:58,630 --> 00:02:03,450
whereas the rectangle was the
basic building block of areas,
39
00:02:03,450 --> 00:02:06,530
the so-called cylinder
will be the basic
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00:02:06,530 --> 00:02:08,130
building block of volumes.
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00:02:08,130 --> 00:02:10,605
Let me take a moment to digress
here and explain to
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00:02:10,605 --> 00:02:12,740
you mathematically what the
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00:02:12,740 --> 00:02:15,390
mathematician calls a cylinder.
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00:02:15,390 --> 00:02:20,160
We start with any closed curve,
say, in a plane, and we
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00:02:20,160 --> 00:02:23,670
then take a line perpendicular
to that plane.
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00:02:23,670 --> 00:02:28,320
And with that line we trace
along the curve.
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00:02:28,320 --> 00:02:31,660
And we then take another plane
parallel to the plane that the
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00:02:31,660 --> 00:02:35,030
curve is in and slice this
thing off someplace.
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00:02:35,030 --> 00:02:41,260
In other words, by definition, a
cylinder has congruent cross
50
00:02:41,260 --> 00:02:43,250
sections all the way through.
51
00:02:43,250 --> 00:02:46,930
And what we're saying is that--
in fact, the familiar
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00:02:46,930 --> 00:02:49,130
form of a cylinder is the
one where the cross
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00:02:49,130 --> 00:02:50,260
section is a circle.
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00:02:50,260 --> 00:02:52,390
That's called the right
circular cylinder.
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00:02:52,390 --> 00:02:55,120
Remember, the volume of a right
circular cylinder is 'pi
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00:02:55,120 --> 00:02:58,230
r squared h', the area
of the cross
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00:02:58,230 --> 00:02:59,860
section times the height.
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00:02:59,860 --> 00:03:01,760
Well, that's the generalization
that we make.
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00:03:01,760 --> 00:03:05,270
In other words, our first
assumption is that for any
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00:03:05,270 --> 00:03:09,080
cylinder the volume of
the cylinder is the
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00:03:09,080 --> 00:03:13,730
cross-sectional area
times the height.
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00:03:13,730 --> 00:03:16,500
Or you could call it the area of
the base times the height,
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00:03:16,500 --> 00:03:19,870
since the cross-sectional area
is the same for all slices.
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OK?
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00:03:21,180 --> 00:03:25,480
The next assumption says, if we
think of volume as meaning
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00:03:25,480 --> 00:03:28,960
the amount of space only in
three dimensions, whereas area
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00:03:28,960 --> 00:03:31,650
means the amount of space in
two dimensions, our next
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00:03:31,650 --> 00:03:34,700
assumption is that if the three
dimensional region 'R'
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00:03:34,700 --> 00:03:37,860
is contained in the three
dimensional region 'S', then
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the volume of 'R' is less than
or equal to the volume of 'S'.
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00:03:42,270 --> 00:03:46,460
And finally, we assume an
analogous result about the
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00:03:46,460 --> 00:03:49,470
area of the whole equals the sum
of the areas of the parts.
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00:03:49,470 --> 00:03:54,310
We assume that if a region is
made up of the union of 'n'
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00:03:54,310 --> 00:03:56,640
regions which do not overlap--
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00:03:56,640 --> 00:03:59,460
notice the union
notation here.
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00:03:59,460 --> 00:04:04,010
The union of 'R sub 1' up to 'R
sub n' and if the 'R's do
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00:04:04,010 --> 00:04:08,730
not overlap, then the volume of
the region 'R' is the sum
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00:04:08,730 --> 00:04:12,860
of the volumes of the
constituent parts.
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00:04:12,860 --> 00:04:15,360
In other words, notice that
except for the fact that
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00:04:15,360 --> 00:04:20,910
cylinder replaces rectangle, the
basic axioms for studying
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00:04:20,910 --> 00:04:24,030
volume are precisely the
same as the axioms
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00:04:24,030 --> 00:04:25,270
for studying area.
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00:04:25,270 --> 00:04:28,370
In particular, then, what this
means is that structurally the
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00:04:28,370 --> 00:04:32,510
same results that we were able
to show for area should follow
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00:04:32,510 --> 00:04:35,810
word for word, essentially,
for volumes.
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00:04:35,810 --> 00:04:38,360
And I thought what we would
do is start with a rather
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00:04:38,360 --> 00:04:39,870
familiar example.
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00:04:39,870 --> 00:04:40,780
You may recall--
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00:04:40,780 --> 00:04:43,030
of course, we've used this
result many times--
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00:04:43,030 --> 00:04:47,110
that the volume of a cylinder
is given by 1/3--
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00:04:47,110 --> 00:04:51,810
that the volume of a cone is
'1/3 pi r squared h' where 'r'
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00:04:51,810 --> 00:04:55,080
is the radius of the base
and 'h' is the height.
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00:04:55,080 --> 00:04:57,800
And you may remember that
in solid geometry as a
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00:04:57,800 --> 00:05:00,780
traditional high school
curriculum went, these results
95
00:05:00,780 --> 00:05:03,890
were given but seldom
if ever proved.
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00:05:03,890 --> 00:05:07,040
So what I thought we would do
now is see how we can use
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00:05:07,040 --> 00:05:10,950
these axioms to arrive at these
results, and to get the
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00:05:10,950 --> 00:05:13,750
spirit of what we've been trying
to do, I will do this
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00:05:13,750 --> 00:05:17,800
by integral calculus at least
first and then by differential
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00:05:17,800 --> 00:05:19,150
calculus later.
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00:05:19,150 --> 00:05:20,960
But the idea is something
like this.
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00:05:20,960 --> 00:05:25,610
To visualize the cone, the
radius of whose base is 'r'
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00:05:25,610 --> 00:05:29,600
and whose height is 'h', we can
think of the straight line
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00:05:29,600 --> 00:05:34,420
that joins the origin to the
point (r, h), this region
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00:05:34,420 --> 00:05:38,370
here, this right triangle.
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00:05:38,370 --> 00:05:41,410
And we can think of that as
being revolved about the
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00:05:41,410 --> 00:05:44,280
x-axis to give the cone.
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00:05:44,280 --> 00:05:47,910
Now when we were dealing with
areas, you may recall that we
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00:05:47,910 --> 00:05:53,060
broke things down into
rectangles that were too big,
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00:05:53,060 --> 00:05:56,460
rectangles that were too small,
and we computed, say,
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00:05:56,460 --> 00:06:00,000
'U sub n' and 'L sub
n', et cetera.
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00:06:00,000 --> 00:06:02,460
We can do the same thing now.
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00:06:02,460 --> 00:06:06,710
What we do is we again
circumscribe rectangles.
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00:06:06,710 --> 00:06:11,230
Now the idea is whatever volume
is traced out by this
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00:06:11,230 --> 00:06:14,960
piece here, whatever volume is
traced out when we rotate this
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00:06:14,960 --> 00:06:20,280
triangle about the x-axis, that
volume will be less than
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00:06:20,280 --> 00:06:24,670
the volume generated by this
particular rectangle, because,
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00:06:24,670 --> 00:06:27,430
you see, the volume that we're
looking for is contained
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00:06:27,430 --> 00:06:31,270
inside the rectangle when we
revolve this particular thing.
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00:06:31,270 --> 00:06:34,070
Now notice that this particular
rectangle when
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00:06:34,070 --> 00:06:37,290
revolved gives me a right
circular cylinder, and we're
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00:06:37,290 --> 00:06:39,750
assuming that we know how to
find the volume of a cylinder.
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00:06:39,750 --> 00:06:42,760
It's the cross sectional
area times the height.
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00:06:42,760 --> 00:06:46,160
Let's focus our attention on
what I call the k-th region
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00:06:46,160 --> 00:06:48,670
here and see what this
thing looks like.
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00:06:48,670 --> 00:06:51,480
First of all, notice that if
we've divided this length,
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00:06:51,480 --> 00:06:55,880
which is 'h' units long into 'n'
equal parts, each of these
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00:06:55,880 --> 00:06:59,430
pieces is 'h' over 'n'.
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00:06:59,430 --> 00:07:03,460
Secondly, notice that the radius
of the base of the
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00:07:03,460 --> 00:07:05,930
cylinder that we're going to
get-- well, let's see.
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00:07:05,930 --> 00:07:08,290
It's going to be this
y-coordinate.
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00:07:08,290 --> 00:07:12,300
Given the x-coordinate, 'y' is
determined by multiplying the
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00:07:12,300 --> 00:07:15,160
x-coordinate by 'r/h'.
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00:07:15,160 --> 00:07:18,300
The x-coordinate is
'kh' over 'n'.
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00:07:18,300 --> 00:07:20,770
I multiply that by 'r/h'.
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00:07:20,770 --> 00:07:23,530
That gives me 'kr' over 'n'.
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00:07:23,530 --> 00:07:25,640
That's the height of this--
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00:07:25,640 --> 00:07:26,900
the radius of the base
of the cylinder that
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00:07:26,900 --> 00:07:28,310
we're going to revolve.
140
00:07:28,310 --> 00:07:30,390
Now what is the volume
of this cylinder?
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00:07:30,390 --> 00:07:35,600
The area of the base is 'pi y
squared', and I'm now going to
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00:07:35,600 --> 00:07:40,040
multiply that by the height,
which is 'h/n'.
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00:07:40,040 --> 00:07:42,830
And if I do that,
I obtain what?
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00:07:42,830 --> 00:07:50,310
The volume of this particular
cylinder is 'pi r squared h'
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00:07:50,310 --> 00:07:53,450
times 'k squared'
over 'n cubed'.
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00:07:53,450 --> 00:07:58,200
And now, if I add up all of
these volumes as 'k' goes from
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00:07:58,200 --> 00:08:04,040
1 to 'n', that will give me a
bunch of stacked cylinders
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00:08:04,040 --> 00:08:08,240
which enclose my cone.
149
00:08:08,240 --> 00:08:10,520
In other words, an answer
that will be too
150
00:08:10,520 --> 00:08:12,210
large will be what?
151
00:08:12,210 --> 00:08:17,610
This sum as 'k' goes from 1 to
'n', notice that this is the
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00:08:17,610 --> 00:08:21,510
only portion that depends
on 'k', so the upper
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00:08:21,510 --> 00:08:22,310
approximation--
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00:08:22,310 --> 00:08:24,390
in other words, the volume
that's too large to be the
155
00:08:24,390 --> 00:08:25,410
right answer--
156
00:08:25,410 --> 00:08:29,180
is 'pi r squared h' over 'n
cubed' times the sum as 'k'
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00:08:29,180 --> 00:08:31,270
goes from 1 to 'n',
'k squared''.
158
00:08:31,270 --> 00:08:35,000
Now you notice I always stick
to problems where we have
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00:08:35,000 --> 00:08:37,530
something fairly simple like
this, because this limit
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00:08:37,530 --> 00:08:40,340
process, as we've mentioned in
the previous lectures, becomes
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00:08:40,340 --> 00:08:43,970
very, very difficult to do in
general, the beauty or one of
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00:08:43,970 --> 00:08:45,990
the beauties of our fundamental
theorem.
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00:08:45,990 --> 00:08:50,870
But the idea is I do know that
this sum is 'n' times 'n + 1'
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00:08:50,870 --> 00:08:54,370
times '2n + 1' over 6.
165
00:08:54,370 --> 00:08:58,730
Now distributing the 'n cubed'
one factor at a time, the way
166
00:08:58,730 --> 00:09:02,970
we have before, I can now write
that this is '1/6 pi r
167
00:09:02,970 --> 00:09:04,260
squared h'.
168
00:09:04,260 --> 00:09:08,000
'n + 1' over 'n'
is '1 + '1/n''.
169
00:09:08,000 --> 00:09:11,840
'2n + 1' over 'n'
is '2 + '1/n''.
170
00:09:11,840 --> 00:09:15,000
And I find that my upper
approximation is given by this
171
00:09:15,000 --> 00:09:16,250
expression.
172
00:09:16,250 --> 00:09:19,860
And if I now take the limit of
'U sub n' as 'n' goes to
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00:09:19,860 --> 00:09:22,500
infinity, this factor
approaches 1.
174
00:09:22,500 --> 00:09:24,370
This factor approaches 2.
175
00:09:24,370 --> 00:09:28,940
Therefore, our entire product
approaches in the limit '1/3
176
00:09:28,940 --> 00:09:31,740
pi r squared h', which
is the familiar
177
00:09:31,740 --> 00:09:34,640
result of solid geometry.
178
00:09:34,640 --> 00:09:37,000
Of course, we've taken a lot
for granted over here.
179
00:09:37,000 --> 00:09:40,040
What we've really proven here
is not that the volume of a
180
00:09:40,040 --> 00:09:42,860
cone is '1/3 pi r squared h'.
181
00:09:42,860 --> 00:09:46,340
What we have proven is at the
limit of 'U sub n' as 'n'
182
00:09:46,340 --> 00:09:49,820
approaches infinity, it's
'1/3 pi r squared h'.
183
00:09:49,820 --> 00:09:52,840
The question that comes up is
how do you know that as these
184
00:09:52,840 --> 00:09:56,960
divisions get small
that the volume--
185
00:09:56,960 --> 00:09:59,980
the upper approximation gets
arbitrarily close to the
186
00:09:59,980 --> 00:10:01,100
correct answer.
187
00:10:01,100 --> 00:10:04,640
And again, notice how we can
reason analogously to what we
188
00:10:04,640 --> 00:10:06,040
did in the case of area.
189
00:10:06,040 --> 00:10:11,820
Namely, what we could do next,
you see, is take the smallest
190
00:10:11,820 --> 00:10:14,310
cylinder that can be
inscribed here.
191
00:10:14,310 --> 00:10:18,090
In other words, this would give
us an approximation which
192
00:10:18,090 --> 00:10:19,960
is too small.
193
00:10:19,960 --> 00:10:27,650
The total error is no more than
the solid generated by
194
00:10:27,650 --> 00:10:31,440
this hatch region revolving
about the x-axis.
195
00:10:31,440 --> 00:10:39,220
Notice, however, that each of
these pieces fits very nicely
196
00:10:39,220 --> 00:10:43,740
in here, so that the total error
between an approximation
197
00:10:43,740 --> 00:10:46,110
which is too big and an
approximation which is too
198
00:10:46,110 --> 00:10:50,430
small is this height,
which is 'r'.
199
00:10:50,430 --> 00:10:52,590
OK?
200
00:10:52,590 --> 00:10:54,540
Let's see, the cross-sectional
area-- this is 'r'.
201
00:10:54,540 --> 00:10:58,210
So 'pi r squared' is the
area of the base.
202
00:10:58,210 --> 00:11:01,370
The height is 'h/n'.
203
00:11:01,370 --> 00:11:05,430
Notice that 'r' and 'h' are
given constants, therefore, as
204
00:11:05,430 --> 00:11:09,200
'n' goes to infinity, the
numerator stays constant.
205
00:11:09,200 --> 00:11:12,650
The denominator goes
to infinity.
206
00:11:12,650 --> 00:11:14,670
The difference goes to 0.
207
00:11:14,670 --> 00:11:18,490
In other words, again, we can
show that 'U sub n' and 'L sub
208
00:11:18,490 --> 00:11:21,160
n' have a common limit.
209
00:11:21,160 --> 00:11:23,770
In fact, we can generalize this
result rather nicely.
210
00:11:23,770 --> 00:11:27,390
211
00:11:27,390 --> 00:11:29,660
Take this drawing to be whatever
you'd like it to be.
212
00:11:29,660 --> 00:11:35,780
I've simply tried to visualize
here a solid region.
213
00:11:35,780 --> 00:11:37,390
This is a 3-dimensional
region.
214
00:11:37,390 --> 00:11:40,870
It has various cross sections.
215
00:11:40,870 --> 00:11:43,980
And I know that as I look at it
in the 'x' direction, the
216
00:11:43,980 --> 00:11:47,510
region begins at 'x' equals
'a' and terminates at 'x'
217
00:11:47,510 --> 00:11:48,520
equals 'b'.
218
00:11:48,520 --> 00:11:51,420
And the question is how can
I find the volume of this
219
00:11:51,420 --> 00:11:57,060
particular region, assuming I
know the cross-sectional area
220
00:11:57,060 --> 00:11:58,420
for any slice?
221
00:11:58,420 --> 00:11:59,770
And the idea, again, is what?
222
00:11:59,770 --> 00:12:04,600
We can slice this solid up into
'n' parts, which I call
223
00:12:04,600 --> 00:12:07,410
'delta V1' up to 'delta Vn'.
224
00:12:07,410 --> 00:12:10,480
The sum of these would be
the true volume that
225
00:12:10,480 --> 00:12:12,020
we're looking for.
226
00:12:12,020 --> 00:12:17,550
Now, I focus my attention on
the k-th piece here, and
227
00:12:17,550 --> 00:12:22,210
again, what I do is I inscribe
and circumscribe cylinders,
228
00:12:22,210 --> 00:12:25,800
one of which is contained
entirely within 'delta V sub
229
00:12:25,800 --> 00:12:29,680
k', and the other of which
surrounds 'delta V sub k'.
230
00:12:29,680 --> 00:12:33,150
In other words, what I do is is
I find the biggest possible
231
00:12:33,150 --> 00:12:36,840
cross-sectional area I have in
this interval, and I denote
232
00:12:36,840 --> 00:12:40,840
the 'x' value at which that
occurs by 'M sub k'.
233
00:12:40,840 --> 00:12:46,190
I find the value of 'x', which
I call 'm sub k', at which I
234
00:12:46,190 --> 00:12:48,970
get the smallest cross-sectional
area.
235
00:12:48,970 --> 00:12:52,160
And therefore, the inscribed
cylinder has
236
00:12:52,160 --> 00:12:55,090
volume given by this.
237
00:12:55,090 --> 00:13:00,300
The circumscribed cylinder has
volume given by this, and the
238
00:13:00,300 --> 00:13:03,690
piece that I'm looking for,
the true volume, is caught
239
00:13:03,690 --> 00:13:05,220
between these two.
240
00:13:05,220 --> 00:13:09,300
Therefore, if I add these up
as 'k' goes from 1 to 'n',
241
00:13:09,300 --> 00:13:13,770
I've caught 'V' between 'U
sub n' and 'L sub n'.
242
00:13:13,770 --> 00:13:18,170
Assuming only that the area is
a continuous function, the
243
00:13:18,170 --> 00:13:22,120
difference between the largest
cross section and the smallest
244
00:13:22,120 --> 00:13:27,670
cross section approaches 0 as
'delta x sub k' approaches 0.
245
00:13:27,670 --> 00:13:30,690
In essence then, the same as
we did before, what we can
246
00:13:30,690 --> 00:13:34,900
show is that as 'n' goes to
infinity, both 'U sub n' and
247
00:13:34,900 --> 00:13:40,840
'L sub n' approach a common
limit, and therefore, the 'V'
248
00:13:40,840 --> 00:13:44,870
being caught between these two
is equal to the common limit.
249
00:13:44,870 --> 00:13:46,390
Now here's the interesting
point.
250
00:13:46,390 --> 00:13:49,530
When we talked about the
definite integral, no one
251
00:13:49,530 --> 00:13:53,250
asked physically what the
function 'f' was or what the
252
00:13:53,250 --> 00:13:57,010
'c sub k's that we're using here
were, only that they be
253
00:13:57,010 --> 00:14:00,170
in the proper interval and 'f'
be a continuous function.
254
00:14:00,170 --> 00:14:04,430
Notice that we're assuming
that 'A' is continuous.
255
00:14:04,430 --> 00:14:07,320
In essence then, by the
definition of the definite
256
00:14:07,320 --> 00:14:09,900
integral, that volume
is just what?
257
00:14:09,900 --> 00:14:12,970
The definite integral from 'a'
to 'b', 'f of x' 'dx'.
258
00:14:12,970 --> 00:14:17,700
In other words, it's this sum
taken in the limit as 'n'
259
00:14:17,700 --> 00:14:20,590
approaches infinity, or another
way of saying that, as
260
00:14:20,590 --> 00:14:25,400
the maximum 'delta X sub
k' approaches 0.
261
00:14:25,400 --> 00:14:27,800
That, by the way, is the
integral calculus approach.
262
00:14:27,800 --> 00:14:30,580
If we want the differential
calculus approach, remember
263
00:14:30,580 --> 00:14:32,610
what we do, we say look it.
264
00:14:32,610 --> 00:14:37,610
The change in volume is less
than or equal to the maximum
265
00:14:37,610 --> 00:14:42,150
cross-sectional area times
'delta X' and greater than or
266
00:14:42,150 --> 00:14:44,890
equal to the minimum
cross-sectional area
267
00:14:44,890 --> 00:14:46,420
times 'delta X'.
268
00:14:46,420 --> 00:14:49,820
Same as we did for area,
you see, we divide
269
00:14:49,820 --> 00:14:51,700
through by 'delta X'.
270
00:14:51,700 --> 00:14:55,830
We have that 'delta V' divided
by 'delta X' is caught between
271
00:14:55,830 --> 00:15:02,970
'A of M', 'A of m', where 'm'
and 'M' are in this interval.
272
00:15:02,970 --> 00:15:08,370
And now you see as 'delta X'
approaches 0, 'm' and 'M' both
273
00:15:08,370 --> 00:15:09,370
approach 'x'.
274
00:15:09,370 --> 00:15:11,860
You see the same procedure
as we had before.
275
00:15:11,860 --> 00:15:14,780
You see what we're saying going
back to this diagram is,
276
00:15:14,780 --> 00:15:22,040
for example, here is a 'delta
X', and 'm' and 'M' are points
277
00:15:22,040 --> 00:15:27,280
in here, and as 'delta X' goes
to 0, 'm' and 'M' both
278
00:15:27,280 --> 00:15:29,630
approach the end point 'x'.
279
00:15:29,630 --> 00:15:34,170
And since 'A' is assumed to be
continuous, if 'M' and 'm'
280
00:15:34,170 --> 00:15:39,710
approach 'x', 'A of M', 'A of
m' approach 'A of x', and we
281
00:15:39,710 --> 00:15:44,980
arrive at, by differential
calculus, that 'dV dx' is 'A
282
00:15:44,980 --> 00:15:50,600
of x', and therefore, 'V' is
again equal to integral ''A of
283
00:15:50,600 --> 00:15:53,500
x' 'dx'' as 'x' goes
from 'a' to 'b'.
284
00:15:53,500 --> 00:15:57,610
Where now by differential
calculus, this means what?
285
00:15:57,610 --> 00:16:05,890
It simply means 'G of b' minus
'G of a', where 'G' is any
286
00:16:05,890 --> 00:16:09,000
function whose derivative
is 'f'.
287
00:16:09,000 --> 00:16:11,580
Now again, I have to go through
this thing rather
288
00:16:11,580 --> 00:16:15,550
hurriedly because I want to
get some examples done.
289
00:16:15,550 --> 00:16:18,530
But what I hope is that we went
slowly enough so that you
290
00:16:18,530 --> 00:16:22,460
can again sense how we're
using integral calculus,
291
00:16:22,460 --> 00:16:25,910
differential calculus, and the
relationship between them.
292
00:16:25,910 --> 00:16:29,220
Let's, at any rate, illustrate
some of these results more
293
00:16:29,220 --> 00:16:31,610
concretely in terms of--
294
00:16:31,610 --> 00:16:36,160
well, first of all, let's talk
about one particular type of
295
00:16:36,160 --> 00:16:39,920
solid, what's called a
solid of revolution.
296
00:16:39,920 --> 00:16:42,970
That's the particular type of
solid where you have a region
297
00:16:42,970 --> 00:16:47,130
in the 'xy' plane, and you take
that region and rotate it
298
00:16:47,130 --> 00:16:51,580
either around the x-axis or the
y-axis, thus generating a
299
00:16:51,580 --> 00:16:54,240
3-dimensional region.
300
00:16:54,240 --> 00:16:57,710
See, in other words, a plane
area is rotated through 360
301
00:16:57,710 --> 00:17:01,500
degrees either with respect to
the x-axis or the y-axis.
302
00:17:01,500 --> 00:17:03,950
I'll consider the x-axis here.
303
00:17:03,950 --> 00:17:06,880
Notice that this is a special
case of what we've just
304
00:17:06,880 --> 00:17:10,339
studied, namely, in this
particular case, if 'y' equals
305
00:17:10,339 --> 00:17:13,710
'f of x' is a continuous
function, notice that every
306
00:17:13,710 --> 00:17:15,910
cross section here, every cross
307
00:17:15,910 --> 00:17:19,240
section will be a circle.
308
00:17:19,240 --> 00:17:22,670
The area of the circle
is 'pi y squared'.
309
00:17:22,670 --> 00:17:24,920
That's 'pi 'f of x' squared'.
310
00:17:24,920 --> 00:17:28,180
And therefore, according to our
fundamental theorem, since
311
00:17:28,180 --> 00:17:29,770
the area is continuous--
312
00:17:29,770 --> 00:17:31,380
and why is the area
continuous?
313
00:17:31,380 --> 00:17:34,280
Well, if 'f' is continuous,
remember the product of
314
00:17:34,280 --> 00:17:36,190
continuous functions
is continuous.
315
00:17:36,190 --> 00:17:39,170
If 'f' is continuous, 'f
squared' is continuous.
316
00:17:39,170 --> 00:17:42,330
So according to our result, the
volume of the region 'R'
317
00:17:42,330 --> 00:17:45,010
is just the integral from
'a' to 'b', 'pi 'f
318
00:17:45,010 --> 00:17:46,660
of x' squared' 'dx'.
319
00:17:46,660 --> 00:17:48,960
And you see, using differential
calculus, all we
320
00:17:48,960 --> 00:17:52,520
need now is a function whose
derivative is 'pi 'f of x'
321
00:17:52,520 --> 00:17:54,890
squared', and we call
that function 'G'.
322
00:17:54,890 --> 00:17:57,770
We compute 'G of b' minus 'G
of a' and that gives us the
323
00:17:57,770 --> 00:18:00,120
volume that we're looking for.
324
00:18:00,120 --> 00:18:02,830
Remember, when we talked about
areas, we mentioned this was
325
00:18:02,830 --> 00:18:04,020
highly specialized.
326
00:18:04,020 --> 00:18:06,550
What if you had a region
like this?
327
00:18:06,550 --> 00:18:09,540
And again, sparing the details,
observe that if we
328
00:18:09,540 --> 00:18:13,290
have a region like this, we can
draw in the lines where
329
00:18:13,290 --> 00:18:15,130
the curve doubles back.
330
00:18:15,130 --> 00:18:21,010
We can now visualize this
as the volume--
331
00:18:21,010 --> 00:18:22,630
the difference of two volumes.
332
00:18:22,630 --> 00:18:26,970
Namely, we can find this volume
and subtract from that
333
00:18:26,970 --> 00:18:30,440
volume this volume.
334
00:18:30,440 --> 00:18:33,590
See, in other words, both of
these have the right form.
335
00:18:33,590 --> 00:18:36,430
And by this difference,
what's left?
336
00:18:36,430 --> 00:18:39,170
The difference of the big volume
minus the small volume
337
00:18:39,170 --> 00:18:41,070
is the volume generated
by 'R'.
338
00:18:41,070 --> 00:18:44,750
And as I say, these are rather
simple details that we can
339
00:18:44,750 --> 00:18:48,640
check out computationally in
terms of exercises, but the
340
00:18:48,640 --> 00:18:50,990
reason I wanted to mention the
solid of revolution is that
341
00:18:50,990 --> 00:18:55,110
not only is this a rather common
and important category,
342
00:18:55,110 --> 00:18:58,250
but it also happens to be the
type of solid that we opened
343
00:18:58,250 --> 00:18:59,450
our program with.
344
00:18:59,450 --> 00:19:03,700
Remember, the cone may
be viewed as what?
345
00:19:03,700 --> 00:19:08,330
The solid generated by a
particular right triangle
346
00:19:08,330 --> 00:19:11,110
being revolved about
the x-axis.
347
00:19:11,110 --> 00:19:13,860
In fact, I thought we could
try that same problem now
348
00:19:13,860 --> 00:19:17,350
doing it by the antiderivative
method.
349
00:19:17,350 --> 00:19:21,030
Namely, we take this particular
region 'R' and
350
00:19:21,030 --> 00:19:22,980
notice now, if we revolve
this about the x-axis--
351
00:19:22,980 --> 00:19:25,520
352
00:19:25,520 --> 00:19:28,080
let's see, the cross-sectional
area will be what?
353
00:19:28,080 --> 00:19:31,470
Well, it's a circle
of radius 'y'.
354
00:19:31,470 --> 00:19:34,430
For a given value of 'x',
'y' is equal to 'r'
355
00:19:34,430 --> 00:19:36,040
times 'x' over 'h'.
356
00:19:36,040 --> 00:19:38,080
See the slope of this
line is 'r/h'.
357
00:19:38,080 --> 00:19:39,920
It passes through
the origin here.
358
00:19:39,920 --> 00:19:43,860
So the cross-sectional area
is 'y squared' times pi.
359
00:19:43,860 --> 00:19:47,950
That's 'pi 'r squared h'
squared' over 'h squared'
360
00:19:47,950 --> 00:19:49,720
times 'x squared'.
361
00:19:49,720 --> 00:19:53,080
And to find that volume,
I simply integrate this
362
00:19:53,080 --> 00:19:56,190
between 0 and 'h'.
363
00:19:56,190 --> 00:19:59,600
Recalling that pi, 'r', and 'h'
are constants, I can take
364
00:19:59,600 --> 00:20:01,900
the constants outside of
the integral sign.
365
00:20:01,900 --> 00:20:05,140
The integral of 'x squared',
meaning what?
366
00:20:05,140 --> 00:20:09,070
The inverse derivative
is '1/3 x cubed'.
367
00:20:09,070 --> 00:20:14,590
If I evaluate that between 0 and
'h', I get '1/3 h cubed'.
368
00:20:14,590 --> 00:20:18,820
The 'h cubed' in the numerator
cancels the 'h squared' in the
369
00:20:18,820 --> 00:20:22,000
denominator, leaving a factor of
'h' in the numerator, and I
370
00:20:22,000 --> 00:20:26,000
wind up, as I saw before, that
the volume of this cone is
371
00:20:26,000 --> 00:20:28,380
'1/3 pi r squared h'.
372
00:20:28,380 --> 00:20:30,840
And this is nice that I get
the same answer as by the
373
00:20:30,840 --> 00:20:33,300
limit method, because according
to the fundamental
374
00:20:33,300 --> 00:20:36,280
theorem, the first fundamental
theorem, this is precisely
375
00:20:36,280 --> 00:20:37,560
what was supposed to happen.
376
00:20:37,560 --> 00:20:40,890
In other words, I can do these
either by limits or by
377
00:20:40,890 --> 00:20:41,680
derivatives.
378
00:20:41,680 --> 00:20:44,430
I want you to see these things
side by side, because in
379
00:20:44,430 --> 00:20:47,120
certain cases, as I've
emphasized in the previous
380
00:20:47,120 --> 00:20:51,140
lectures, there will be times
when we cannot, by
381
00:20:51,140 --> 00:20:55,000
differential calculus, find a
function 'G' whose derivative
382
00:20:55,000 --> 00:20:57,350
is equal to a given
function 'f of x'.
383
00:20:57,350 --> 00:20:59,940
But enough about that
for the time being.
384
00:20:59,940 --> 00:21:03,620
The next question that comes up
gives us a review of what
385
00:21:03,620 --> 00:21:05,140
happens with inverse
functions.
386
00:21:05,140 --> 00:21:07,690
It's a rather interesting
type of situation.
387
00:21:07,690 --> 00:21:10,060
It's called the method of
cylindrical shells and it's
388
00:21:10,060 --> 00:21:11,930
motivated by the following.
389
00:21:11,930 --> 00:21:15,210
Let's suppose again we're given
a very nice region 'R'.
390
00:21:15,210 --> 00:21:16,550
What do I mean by very nice?
391
00:21:16,550 --> 00:21:19,080
Well, to simplify the
computation, even though it
392
00:21:19,080 --> 00:21:23,670
doesn't change the theory at
all, I'm assuming that 'y'
393
00:21:23,670 --> 00:21:26,730
equals 'f of x' is an
increasing curve.
394
00:21:26,730 --> 00:21:29,190
In other words, I'm even
assuming that we have a
395
00:21:29,190 --> 00:21:32,200
one-to-one function here.
396
00:21:32,200 --> 00:21:36,220
Now the idea is here's this
nice region and instead of
397
00:21:36,220 --> 00:21:38,950
revolving this about the
x-axis, I would like to
398
00:21:38,950 --> 00:21:41,880
revolve it about the y-axis.
399
00:21:41,880 --> 00:21:47,000
Now you see, to use the method
of revolution here, to revolve
400
00:21:47,000 --> 00:21:50,470
this about the y-axis,
essentially what I do is I
401
00:21:50,470 --> 00:21:55,060
pick a washer-shaped
region, you see?
402
00:21:55,060 --> 00:21:59,265
I have to compute the volume
generated by the 'y' part.
403
00:21:59,265 --> 00:22:01,810
See, in other words, I do this
as two separate parts.
404
00:22:01,810 --> 00:22:06,740
I find the volume of the big
piece minus the volume of the
405
00:22:06,740 --> 00:22:10,330
small piece, and what's
left is the volume
406
00:22:10,330 --> 00:22:11,680
generated by 'R'.
407
00:22:11,680 --> 00:22:13,970
But notice a rather difficult
computational
408
00:22:13,970 --> 00:22:15,510
thing occurs here.
409
00:22:15,510 --> 00:22:20,160
Namely, notice that this
length here has to be
410
00:22:20,160 --> 00:22:25,010
expressed as 'x' goes from one
value to another value.
411
00:22:25,010 --> 00:22:29,510
Now, you see, if this is 'b',
and this is 'a', you see,
412
00:22:29,510 --> 00:22:31,170
notice what's happening
here, how our
413
00:22:31,170 --> 00:22:32,770
strips are being chosen.
414
00:22:32,770 --> 00:22:37,360
You see, for a given strip, the
final 'x' value is 'b',
415
00:22:37,360 --> 00:22:39,590
but what is the initial
'x' value?
416
00:22:39,590 --> 00:22:41,840
See, down here the 'x'
value is 'a', but
417
00:22:41,840 --> 00:22:43,260
what happens up here?
418
00:22:43,260 --> 00:22:45,670
In other words, how do you find
what the 'x' value is for
419
00:22:45,670 --> 00:22:47,160
a given value of 'y' here?
420
00:22:47,160 --> 00:22:52,660
Well, you see what you must do
is invert the relationship.
421
00:22:52,660 --> 00:22:54,040
Now even though I've
picked a case
422
00:22:54,040 --> 00:22:55,610
where the inverse exists--
423
00:22:55,610 --> 00:22:57,570
see, this is a one-to-one
function--
424
00:22:57,570 --> 00:23:00,910
we've already had ample examples
in which we've shown
425
00:23:00,910 --> 00:23:04,040
that computationally it's
extremely difficult if it's
426
00:23:04,040 --> 00:23:10,010
even possible to explicitly
perform the inversion.
427
00:23:10,010 --> 00:23:11,860
And this is where the method
of cylindrical
428
00:23:11,860 --> 00:23:13,290
shells comes from.
429
00:23:13,290 --> 00:23:16,230
Essentially what the method of
cylindrical shells says is
430
00:23:16,230 --> 00:23:19,080
wouldn't it have been nice if
we chose our generating
431
00:23:19,080 --> 00:23:20,920
element to be this way?
432
00:23:20,920 --> 00:23:23,750
In other words, what we say
is look at this piece
433
00:23:23,750 --> 00:23:25,240
of area over here.
434
00:23:25,240 --> 00:23:30,980
One way of visualizing this
solid being rotated is to
435
00:23:30,980 --> 00:23:35,390
think of this particular region
being rotated about the
436
00:23:35,390 --> 00:23:38,990
x-axis, and it generates
a certain volume.
437
00:23:38,990 --> 00:23:41,610
By the way, what volume
will it generate?
438
00:23:41,610 --> 00:23:45,090
The volume that it will generate
will be less than the
439
00:23:45,090 --> 00:23:49,310
volume that this rectangle
generates but greater than the
440
00:23:49,310 --> 00:23:52,660
volume that this rectangle
here generates.
441
00:23:52,660 --> 00:23:54,800
Now what is the volume
generated
442
00:23:54,800 --> 00:23:56,410
by the large rectangle?
443
00:23:56,410 --> 00:23:59,160
And, by the way, notice that I
mean by the volume generated
444
00:23:59,160 --> 00:24:03,570
by the rectangle think of this
as being a slab of a certain
445
00:24:03,570 --> 00:24:06,530
amount of material and I
rotate that slab around
446
00:24:06,530 --> 00:24:08,080
through 360 degrees.
447
00:24:08,080 --> 00:24:11,420
The volume I'm thinking of is
the volume of the material in
448
00:24:11,420 --> 00:24:13,930
that slab, not the material
that's enclosed.
449
00:24:13,930 --> 00:24:15,490
It would be like, if
you're thinking in
450
00:24:15,490 --> 00:24:17,910
terms of a tin can.
451
00:24:17,910 --> 00:24:21,150
I'm not thinking of the volume
enclosed by the tin can.
452
00:24:21,150 --> 00:24:23,850
I'm thinking of the volume
of the tin itself that
453
00:24:23,850 --> 00:24:24,930
makes up the can.
454
00:24:24,930 --> 00:24:28,410
Well, you see again, to go
through this thing as rapidly
455
00:24:28,410 --> 00:24:32,010
as possible but still hitting
the main points, you see,
456
00:24:32,010 --> 00:24:34,920
notice that the volume that I'm
looking for, what is the
457
00:24:34,920 --> 00:24:38,050
volume that's cut out by
this big rectangle?
458
00:24:38,050 --> 00:24:44,240
Well, notice that the area of
the base from-- if I look at
459
00:24:44,240 --> 00:24:51,170
this as being this cylinder
minus this cylinder, the
460
00:24:51,170 --> 00:24:55,560
volume of the big cylinder is
pi times ''x + delta X'
461
00:24:55,560 --> 00:24:58,560
squared' times the height
here, which is 'f
462
00:24:58,560 --> 00:25:01,090
of 'x + delta X'.
463
00:25:01,090 --> 00:25:04,300
And the volume of the
hollow part from
464
00:25:04,300 --> 00:25:06,090
here to here is what?
465
00:25:06,090 --> 00:25:07,780
It's 'pi x squared'--
466
00:25:07,780 --> 00:25:09,420
that's the radius
of the base--
467
00:25:09,420 --> 00:25:12,850
times the height, which is still
'f of 'x + delta X''.
468
00:25:12,850 --> 00:25:19,550
In other words, 'delta V' is
bounded above by this volume.
469
00:25:19,550 --> 00:25:22,150
In other words, as messy as this
looks, that's only what?
470
00:25:22,150 --> 00:25:25,610
That's the volume of the region
471
00:25:25,610 --> 00:25:28,370
generated by this big rectangle.
472
00:25:28,370 --> 00:25:30,970
If we take the smallest
rectangle, namely the one
473
00:25:30,970 --> 00:25:34,660
that's inscribed inside this
region, we get the same
474
00:25:34,660 --> 00:25:38,230
results, except that the height
is now replaced by 'f
475
00:25:38,230 --> 00:25:42,300
of x' rather than by 'f
of 'x + delta X''.
476
00:25:42,300 --> 00:25:45,670
In other words, we catch
'delta V' between two
477
00:25:45,670 --> 00:25:48,820
expressions involving 'x'.
478
00:25:48,820 --> 00:25:51,210
By the way, notice how the
bracketed expression
479
00:25:51,210 --> 00:25:55,370
simplifies the 'pi x squared'
here cancels with the 'pi x
480
00:25:55,370 --> 00:25:59,700
squared' here leaving inside the
parentheses just '2x delta
481
00:25:59,700 --> 00:26:02,130
X' plus 'delta X squared'.
482
00:26:02,130 --> 00:26:04,900
In other words, simplifying
this thing, I can now show
483
00:26:04,900 --> 00:26:09,910
that 'delta V' is caught between
these two expressions
484
00:26:09,910 --> 00:26:13,520
now, this expression here, which
is too big, and this
485
00:26:13,520 --> 00:26:16,490
expression here, which
is too small.
486
00:26:16,490 --> 00:26:19,470
Now I divide by 'delta X'.
487
00:26:19,470 --> 00:26:23,470
The usual procedure to find
'dV dx', it's 'delta V'
488
00:26:23,470 --> 00:26:24,960
divided by 'delta X'.
489
00:26:24,960 --> 00:26:28,140
Then I will take the limit as
'delta X' approaches 0.
490
00:26:28,140 --> 00:26:30,240
You see, so I divide through
by 'delta X'.
491
00:26:30,240 --> 00:26:33,040
We're assuming, of course,
that 'delta X' is not 0.
492
00:26:33,040 --> 00:26:36,660
That's what the limit means
as 'delta X' approaches 0.
493
00:26:36,660 --> 00:26:40,330
You see, it's not zero, but it
gets arbitrarily close to 0.
494
00:26:40,330 --> 00:26:44,680
Notice then that my 'delta V'
divided by 'delta X' is caught
495
00:26:44,680 --> 00:26:51,190
between pi times '2x + delta X'
times 'f of 'x + delta X'',
496
00:26:51,190 --> 00:26:56,480
and pi times '2x + delta
X' times 'f of x'.
497
00:26:56,480 --> 00:26:59,350
And I now let 'delta
X' approach 0.
498
00:26:59,350 --> 00:27:00,580
And here's the key point.
499
00:27:00,580 --> 00:27:04,470
As 'delta X' approaches 0,
notice that the left hand side
500
00:27:04,470 --> 00:27:09,540
becomes '2 pi x'
times 'f of x'.
501
00:27:09,540 --> 00:27:13,140
Notice also what happens
to the right hand side.
502
00:27:13,140 --> 00:27:17,200
This factor, as 'delta X'
approaches 0, becomes '2x'.
503
00:27:17,200 --> 00:27:22,220
And because 'f' is continuous,
as 'delta X' approaches 0, 'f
504
00:27:22,220 --> 00:27:26,370
of 'x + delta X'' approaches
'f of x'.
505
00:27:26,370 --> 00:27:29,510
In other words, then, in the
limit, as 'delta X' approaches
506
00:27:29,510 --> 00:27:36,560
0, I have that 'dV dx' on the
one hand can't be any greater
507
00:27:36,560 --> 00:27:38,780
than '2 pi x' times 'f of x'.
508
00:27:38,780 --> 00:27:41,640
On the other hand, it can't
be any less than '2 pi
509
00:27:41,640 --> 00:27:43,410
x' times 'f of x'.
510
00:27:43,410 --> 00:27:48,770
Consequently, it must equal
'2 pi x' times 'f of x'.
511
00:27:48,770 --> 00:27:52,600
Therefore, if this is 'dV dx',
then 'V' itself is the
512
00:27:52,600 --> 00:27:57,830
integral of this thing evaluated
between 'a' and 'b',
513
00:27:57,830 --> 00:27:59,620
because that's where we're
adding these things up from.
514
00:27:59,620 --> 00:28:04,550
In other words, if we're using
differential calculus, this is
515
00:28:04,550 --> 00:28:10,230
'G of b' minus 'G of a' where
'G prime' equals 'f'.
516
00:28:10,230 --> 00:28:13,090
If we're using integral
calculus, we've found the 'U
517
00:28:13,090 --> 00:28:17,510
sub n' and 'L sub n' and we've
caught 'V' between 'U sub n'
518
00:28:17,510 --> 00:28:19,220
and 'L sub n'.
519
00:28:19,220 --> 00:28:23,180
But, in any event, what we've
shown rigorously now is that
520
00:28:23,180 --> 00:28:25,500
by the cylindrical shell
method-- and we'll illustrate
521
00:28:25,500 --> 00:28:28,900
these with examples to finish
off today's lesson--
522
00:28:28,900 --> 00:28:32,500
that the volume is given
by integral from 'a'
523
00:28:32,500 --> 00:28:34,830
to 'b' '2 pi x'--
524
00:28:34,830 --> 00:28:37,120
and let me just replace
'f of x' by 'y' to
525
00:28:37,120 --> 00:28:38,900
make my diagram simpler--
526
00:28:38,900 --> 00:28:40,470
times 'dx'.
527
00:28:40,470 --> 00:28:42,940
And if you want to think of
this in what I call the
528
00:28:42,940 --> 00:28:45,690
traditional engineering point
of view where you think of a
529
00:28:45,690 --> 00:28:49,600
thin rectangle generating a
volume, what we're saying is
530
00:28:49,600 --> 00:28:53,760
if you think of a little thin
piece like this being revolved
531
00:28:53,760 --> 00:28:57,760
to generate, say, some material
in a tin can, notice
532
00:28:57,760 --> 00:29:01,310
that the amount of material
in here will be what?
533
00:29:01,310 --> 00:29:06,030
Well, when you unroll
this thing--
534
00:29:06,030 --> 00:29:07,350
see, this thing sort
of like this.
535
00:29:07,350 --> 00:29:12,600
When you unroll this thing,
the radius is 'x', so the
536
00:29:12,600 --> 00:29:16,300
circumference when you unroll
it will be '2 pi x'.
537
00:29:16,300 --> 00:29:17,870
The height is 'y'.
538
00:29:17,870 --> 00:29:21,940
So the cross-sectional area
is '2 pi x y' and the
539
00:29:21,940 --> 00:29:23,860
thickness is 'dx'.
540
00:29:23,860 --> 00:29:28,550
So if I multiply that by dx,
that gives me the volume
541
00:29:28,550 --> 00:29:29,990
generated by this piece.
542
00:29:29,990 --> 00:29:32,270
And then in the proud tradition
of the sigma
543
00:29:32,270 --> 00:29:35,780
notation, which I'll come back
to in the next lecture to show
544
00:29:35,780 --> 00:29:37,700
how dangerous this really
is, but the
545
00:29:37,700 --> 00:29:39,120
shortcut method is what?
546
00:29:39,120 --> 00:29:42,480
Add up all of these
contributions as 'x' goes from
547
00:29:42,480 --> 00:29:45,370
'a' to 'b'.
548
00:29:45,370 --> 00:29:47,950
At any rate, that's called the
method of cylindrical shells.
549
00:29:47,950 --> 00:29:51,230
Essentially, one uses
cylindrical shells when we
550
00:29:51,230 --> 00:29:54,000
think of a generating element
being parallel
551
00:29:54,000 --> 00:29:55,700
to the axis of rotation.
552
00:29:55,700 --> 00:29:58,790
We use revolution, when
it's perpendicular
553
00:29:58,790 --> 00:30:00,220
to the axis of rotation.
554
00:30:00,220 --> 00:30:04,530
Which of the two is easier
depends on the particular
555
00:30:04,530 --> 00:30:07,830
computational technique
necessitated by the
556
00:30:07,830 --> 00:30:10,930
relationship between the
variables in the problem.
557
00:30:10,930 --> 00:30:14,050
Well, at any rate, let's do
a couple of examples.
558
00:30:14,050 --> 00:30:17,340
The first example I'd like to do
is to take that same region
559
00:30:17,340 --> 00:30:21,680
'R', namely, the right
triangle whose
560
00:30:21,680 --> 00:30:24,000
legs are 'r' and 'h'.
561
00:30:24,000 --> 00:30:28,150
We've already solved this
problem of finding the volume
562
00:30:28,150 --> 00:30:30,620
when we rotate this
about the x-axis.
563
00:30:30,620 --> 00:30:33,830
What I'd now like to do is see
what volume is generated by
564
00:30:33,830 --> 00:30:36,990
this as I rotate it
about the y-axis.
565
00:30:36,990 --> 00:30:40,320
And again, I find that I can
do this problem in several
566
00:30:40,320 --> 00:30:43,580
ways, but I thought it was an
easy enough problem to do by
567
00:30:43,580 --> 00:30:46,660
cylindrical shells, because, as
we so often do, I thought
568
00:30:46,660 --> 00:30:49,240
the first problem that we do by
cylindrical shells should
569
00:30:49,240 --> 00:30:51,650
be one that we can check
by another method.
570
00:30:51,650 --> 00:30:55,260
But at any rate, using
cylindrical shells, let's see
571
00:30:55,260 --> 00:30:57,330
what happens over here.
572
00:30:57,330 --> 00:30:58,530
The volume is what?
573
00:30:58,530 --> 00:31:05,020
It's the integral from 0 to
'h', '2 pi x' times this
574
00:31:05,020 --> 00:31:09,160
height, which is 'rx'
over 'h' integrated
575
00:31:09,160 --> 00:31:10,790
with respect to 'x'.
576
00:31:10,790 --> 00:31:13,900
That's just '2 pi r' over
'h'-- we can take that
577
00:31:13,900 --> 00:31:15,880
outside, because that's
a constant factor--
578
00:31:15,880 --> 00:31:17,980
integral 'x squared dx'.
579
00:31:17,980 --> 00:31:20,930
The integral of 'x squared'
is '1/3 x cubed'.
580
00:31:20,930 --> 00:31:23,540
We evaluate that between
0 and 'h'.
581
00:31:23,540 --> 00:31:25,690
That gives us '1/3 h cubed'.
582
00:31:25,690 --> 00:31:28,760
We cancel the 'h' in the
denominator with one of the
583
00:31:28,760 --> 00:31:31,740
'h's in the numerator, and we
find that the volume that's
584
00:31:31,740 --> 00:31:37,220
generated is '2/3 pi r h
squared', not 'r squared h',
585
00:31:37,220 --> 00:31:40,735
'r h squared', not 1/3, 2/3.
586
00:31:40,735 --> 00:31:43,910
Remember, by the way, what
this thing looks like.
587
00:31:43,910 --> 00:31:45,290
I think you can visualize
this.
588
00:31:45,290 --> 00:31:49,700
This is a cylinder with
a cone cut out of it.
589
00:31:49,700 --> 00:31:52,190
See, in other words, if this
thing had been solid, we'd
590
00:31:52,190 --> 00:31:55,340
have called it a right circular
cylinder, and then
591
00:31:55,340 --> 00:31:59,320
what's missing is the cone
shaped region over here.
592
00:31:59,320 --> 00:32:01,530
In fact, that's how
we can check this.
593
00:32:01,530 --> 00:32:05,090
See, what would the
volume be that's
594
00:32:05,090 --> 00:32:07,300
generated by this rectangle?
595
00:32:07,300 --> 00:32:10,990
This would be a cylinder the
radius of whose base is 'h',
596
00:32:10,990 --> 00:32:14,540
whose height is 'r', and the
volume of that cylinder is 'pi
597
00:32:14,540 --> 00:32:16,230
h squared r'.
598
00:32:16,230 --> 00:32:19,260
The cone that's missing, the
cone that was cut out of this
599
00:32:19,260 --> 00:32:22,920
thing, has the radius of a space
equal to 'h' and its
600
00:32:22,920 --> 00:32:26,510
height equal to 'r', so
its volume is '1/3
601
00:32:26,510 --> 00:32:29,090
pi h squared r'.
602
00:32:29,090 --> 00:32:31,420
And, therefore, the volume
that's left when we subtract
603
00:32:31,420 --> 00:32:35,210
this off is '2/3 pi h
squared r', which
604
00:32:35,210 --> 00:32:37,640
does check with this.
605
00:32:37,640 --> 00:32:41,630
By the way, just as an aside,
notice that the region 'R'
606
00:32:41,630 --> 00:32:45,020
generates a different volume
if we rotate it about the
607
00:32:45,020 --> 00:32:49,620
y-axis than it did if we rotate
it about the x-axis.
608
00:32:49,620 --> 00:32:53,980
Numerically, what we're saying
is that we just found that the
609
00:32:53,980 --> 00:32:57,230
volume when rotated around
the y-axis is '2/3
610
00:32:57,230 --> 00:32:58,940
pi h squared r'.
611
00:32:58,940 --> 00:33:02,790
We know that when we revolve
that about the x-axis, it's
612
00:33:02,790 --> 00:33:05,470
'1/3 pi r squared h',
and these two
613
00:33:05,470 --> 00:33:07,580
expressions are not identical.
614
00:33:07,580 --> 00:33:12,320
In fact, if we divide both sides
by 'pi r h' over 3, we
615
00:33:12,320 --> 00:33:16,830
find that equality holds only
if we have that highly
616
00:33:16,830 --> 00:33:19,540
specialized case that '2h'
equals 'r', which is not
617
00:33:19,540 --> 00:33:20,710
really important.
618
00:33:20,710 --> 00:33:22,550
I just threw that
in as an aside.
619
00:33:22,550 --> 00:33:25,830
But I do want you to notice that
the same area, of course,
620
00:33:25,830 --> 00:33:29,050
generates different volumes
depending on what you rotate
621
00:33:29,050 --> 00:33:30,650
it with respect to.
622
00:33:30,650 --> 00:33:33,270
Well, at any rate, at least
this was a problem that we
623
00:33:33,270 --> 00:33:34,600
could check by another method.
624
00:33:34,600 --> 00:33:37,350
Let me just use cylindrical
shells for a problem which is
625
00:33:37,350 --> 00:33:40,130
slightly tougher but one that
can still be checked by
626
00:33:40,130 --> 00:33:41,420
another method.
627
00:33:41,420 --> 00:33:43,330
Let's take the following
region.
628
00:33:43,330 --> 00:33:47,620
Let's take the curve 'y'
equals '2x - x squared'
629
00:33:47,620 --> 00:33:51,340
between 'x' equals 0
and 'x' equals 2.
630
00:33:51,340 --> 00:33:55,250
Leaving the details as a rather
trivial exercise, it is
631
00:33:55,250 --> 00:33:59,710
not difficult to see that this
is the parabola that peaks at
632
00:33:59,710 --> 00:34:06,270
1, 1 and crosses the x-axis at
'x' equals 0 and 'x' equals 2.
633
00:34:06,270 --> 00:34:10,310
If I now want to compute this
volume as I rotate the region
634
00:34:10,310 --> 00:34:13,120
'R' about the y-axis--
635
00:34:13,120 --> 00:34:15,469
see, I'm going to rotate
this about the y-axis.
636
00:34:15,469 --> 00:34:18,320
I want to find out what volume
is generated by this region
637
00:34:18,320 --> 00:34:19,699
'R' in this case.
638
00:34:19,699 --> 00:34:25,530
Remember I can use either
cylindrical shells or I can
639
00:34:25,530 --> 00:34:29,780
use revolution here.
640
00:34:29,780 --> 00:34:32,270
Notice the problem I'm
going to be in.
641
00:34:32,270 --> 00:34:35,159
Notice that this particular
function is single valued but
642
00:34:35,159 --> 00:34:36,239
not one-to-one.
643
00:34:36,239 --> 00:34:38,830
When I try to find these two
'x' values I'm going to run
644
00:34:38,830 --> 00:34:40,060
into multi-values.
645
00:34:40,060 --> 00:34:41,489
I'm going to have to invert.
646
00:34:41,489 --> 00:34:44,770
All sorts of computational
skills are going to
647
00:34:44,770 --> 00:34:46,000
come into play here.
648
00:34:46,000 --> 00:34:49,060
On the other hand, if I take my
generating element parallel
649
00:34:49,060 --> 00:34:52,510
to the y-axis, I have a very
simple expression for this,
650
00:34:52,510 --> 00:34:55,960
and now that indicates what?
651
00:34:55,960 --> 00:34:59,320
By the method of cylindrical
shells, this will be the
652
00:34:59,320 --> 00:35:02,210
integral from 0 to 2.
653
00:35:02,210 --> 00:35:05,930
The generating arm is 'x', so
that cuts out '2 pi x'.
654
00:35:05,930 --> 00:35:09,470
The height is 'y', which
is '2x - x squared'.
655
00:35:09,470 --> 00:35:10,850
The thickness is 'dx'.
656
00:35:10,850 --> 00:35:13,240
In other words, mechanically
I must evaluate
657
00:35:13,240 --> 00:35:15,230
this particular integral.
658
00:35:15,230 --> 00:35:16,000
OK?
659
00:35:16,000 --> 00:35:19,580
At any rate, it's factoring
out the 2 pi and just
660
00:35:19,580 --> 00:35:22,090
observing that the
integral of '2 x
661
00:35:22,090 --> 00:35:24,040
squared' is '2/3 x cubed'.
662
00:35:24,040 --> 00:35:25,340
The integral of 'x'
to the fourth is
663
00:35:25,340 --> 00:35:27,070
'1/4 'x to the 4th''.
664
00:35:27,070 --> 00:35:32,190
Evaluating that between
0 and 2, I get 2 pi
665
00:35:32,190 --> 00:35:35,490
times 16/3 minus 4.
666
00:35:35,490 --> 00:35:37,300
The lower limit is 0 here.
667
00:35:37,300 --> 00:35:42,330
This just comes out to be 16
minus 12, 4/3 times 2 pi.
668
00:35:42,330 --> 00:35:44,530
That's 8 pi over 3.
669
00:35:44,530 --> 00:35:48,150
By the way, before I go any
further with this, let's make
670
00:35:48,150 --> 00:35:50,350
the interesting observation.
671
00:35:50,350 --> 00:35:53,480
See, 8 pi over 3 is the volume
generated by this whole thing
672
00:35:53,480 --> 00:35:57,140
being revolved about
the y-axis.
673
00:35:57,140 --> 00:36:00,340
If I'd drawn in this line, which
is a line of symmetry,
674
00:36:00,340 --> 00:36:02,980
notice that these two
areas are congruent.
675
00:36:02,980 --> 00:36:04,780
These two regions
are congruent.
676
00:36:04,780 --> 00:36:08,710
However, it's also interesting
to observe that the volume
677
00:36:08,710 --> 00:36:12,700
generated by this piece as you
revolve it about the y-axis is
678
00:36:12,700 --> 00:36:15,390
not twice the volume generated
by this piece.
679
00:36:15,390 --> 00:36:18,640
680
00:36:18,640 --> 00:36:21,190
See, notice that the integral
from 0 to 1-- in other words,
681
00:36:21,190 --> 00:36:23,940
if we just took this region
here, integrated this from 0
682
00:36:23,940 --> 00:36:32,880
to 1, we would get 5/6, 2/3
minus 1/4 times 2 pi, 5/6.
683
00:36:32,880 --> 00:36:36,920
And if we double that,
we would get 5/3.
684
00:36:36,920 --> 00:36:39,450
In other words, this
area is 5/6.
685
00:36:39,450 --> 00:36:44,470
Double it would be
5 pi over 6.
686
00:36:44,470 --> 00:36:48,880
Double it would be 5 pi over 3,
and 5 pi over 3 is not the
687
00:36:48,880 --> 00:36:51,330
same as 8 pi over 3.
688
00:36:51,330 --> 00:36:54,060
The thing to keep in mind here
is notice how the distance
689
00:36:54,060 --> 00:36:55,070
comes in again.
690
00:36:55,070 --> 00:37:01,530
You see, for example, these two
lines here are symmetric
691
00:37:01,530 --> 00:37:05,110
with respect to the
line 'x' equals 1.
692
00:37:05,110 --> 00:37:09,110
But notice that this generates a
much larger volume than this
693
00:37:09,110 --> 00:37:11,720
because its generating
arm is longer.
694
00:37:11,720 --> 00:37:13,200
It's further away.
695
00:37:13,200 --> 00:37:15,330
But at any rate, that's
just an aside.
696
00:37:15,330 --> 00:37:17,790
Notice how by the method of
cylindrical shells, we
697
00:37:17,790 --> 00:37:20,620
determine the volume
is 8 pi over 3.
698
00:37:20,620 --> 00:37:24,620
Suppose we'd wanted to do this
by the solid method-- the
699
00:37:24,620 --> 00:37:26,470
solid revolution method.
700
00:37:26,470 --> 00:37:28,850
Notice that we would first
have to invert this
701
00:37:28,850 --> 00:37:29,880
relationship.
702
00:37:29,880 --> 00:37:33,260
We would first have to solve
for 'x' in terms of 'y'.
703
00:37:33,260 --> 00:37:37,320
Notice that 'y' equals '2x - x
squared' is the same as saying
704
00:37:37,320 --> 00:37:41,140
that ''x squared'
- 2x + y' is 0.
705
00:37:41,140 --> 00:37:44,680
Using the quadratic formula, we
can solve for 'x', and we
706
00:37:44,680 --> 00:37:49,430
now find that 'x' is 1 plus the
square root of '1 - y' or
707
00:37:49,430 --> 00:37:52,800
1 minus the square
root of '1 - y'.
708
00:37:52,800 --> 00:37:57,050
What that means, by the way,
geometrically, is simply this.
709
00:37:57,050 --> 00:38:01,720
For a given value of 'y', there
are two values of 'x'
710
00:38:01,720 --> 00:38:06,340
located symmetrically with
respect to 'x' equals 1.
711
00:38:06,340 --> 00:38:08,200
See, they're on symmetrical
portions.
712
00:38:08,200 --> 00:38:09,920
Well, this doesn't make
that much difference.
713
00:38:09,920 --> 00:38:11,300
The thing now is what?
714
00:38:11,300 --> 00:38:13,680
What is my cross-sectional
area?
715
00:38:13,680 --> 00:38:19,130
My cross-sectional area is pi
times this length squared
716
00:38:19,130 --> 00:38:22,680
minus pi times this
length squared.
717
00:38:22,680 --> 00:38:26,280
That's pi times 1 plus the
square root of ''1 - y'
718
00:38:26,280 --> 00:38:30,280
squared' minus pi times the
square root of 1 minus the
719
00:38:30,280 --> 00:38:32,670
square root of ''1
- y' squared'.
720
00:38:32,670 --> 00:38:35,790
When I square this and subtract,
all but the middle
721
00:38:35,790 --> 00:38:37,180
term drops out.
722
00:38:37,180 --> 00:38:40,640
In other words, I have twice
the square root of '1 - y'
723
00:38:40,640 --> 00:38:44,210
here minus twice the square
root of '1 - y' here.
724
00:38:44,210 --> 00:38:48,610
When I subtract, I get 4 times
the square root of '1 - y'.
725
00:38:48,610 --> 00:38:50,560
I multiply that by pi.
726
00:38:50,560 --> 00:38:53,180
That's my cross-sectional
area.
727
00:38:53,180 --> 00:38:56,260
And now to find the volume, I
just integrate that as 'y'
728
00:38:56,260 --> 00:38:58,800
goes from 0 to 1.
729
00:38:58,800 --> 00:39:02,100
You see, and if I carry out
this integration, noticing
730
00:39:02,100 --> 00:39:06,620
that the integral of ''1 - y'
to the 1/2' is minus 2/3.
731
00:39:06,620 --> 00:39:08,800
Remember, the derivative of
'1 - y' with respect to
732
00:39:08,800 --> 00:39:11,610
'y' is minus 1.
733
00:39:11,610 --> 00:39:13,390
''1 - y' to the 3/2'.
734
00:39:13,390 --> 00:39:15,510
Evaluate that between 0 and 1.
735
00:39:15,510 --> 00:39:17,220
The upper limit gives me 0.
736
00:39:17,220 --> 00:39:19,370
The lower limit is minus 2/3.
737
00:39:19,370 --> 00:39:20,550
I subtract the lower limit.
738
00:39:20,550 --> 00:39:21,800
It gives me 2/3.
739
00:39:21,800 --> 00:39:25,860
4 pi times 2/3 is 8 pi
over 3, the same
740
00:39:25,860 --> 00:39:27,940
answer as I got before.
741
00:39:27,940 --> 00:39:30,760
Notice, by the way, that
this was messy, but we
742
00:39:30,760 --> 00:39:31,770
could handle it.
743
00:39:31,770 --> 00:39:33,580
If this had been
much tougher--
744
00:39:33,580 --> 00:39:37,680
say a 6 over here or something
like that instead of a 2--
745
00:39:37,680 --> 00:39:41,010
how would we have solved for
'x' in terms of 'y'?
746
00:39:41,010 --> 00:39:43,370
You see, in other words, this
would have been a case where
747
00:39:43,370 --> 00:39:46,250
the shell method would have been
necessitated because of
748
00:39:46,250 --> 00:39:48,780
the impossibility
of the algebra.
749
00:39:48,780 --> 00:39:51,660
But at any rate, we have plenty
of opportunity to
750
00:39:51,660 --> 00:39:55,550
illustrate that in terms of
exercises and supplementary
751
00:39:55,550 --> 00:39:58,100
notes and reading material
and what have you.
752
00:39:58,100 --> 00:40:00,330
That is actually the
easiest part.
753
00:40:00,330 --> 00:40:02,840
The hard part is to understand
the significance of what's
754
00:40:02,840 --> 00:40:07,170
going on, so I thought that to
summarize today's lecture,
755
00:40:07,170 --> 00:40:10,620
let's keep in mind that whether
you call it area or
756
00:40:10,620 --> 00:40:14,530
whether you call it volume or
whether you call it distance
757
00:40:14,530 --> 00:40:18,870
traveled in velocity, the fact
remains that if 'f' is a
758
00:40:18,870 --> 00:40:21,800
function continuous on the
closed interval from 'a' to
759
00:40:21,800 --> 00:40:27,930
'b', and we partition that
interval into 'n' parts, and
760
00:40:27,930 --> 00:40:32,270
we form the sum as 'k' goes from
1 to 'n', 'f of 'c sub
761
00:40:32,270 --> 00:40:37,150
k'' times 'delta x sub k', where
'c sub k' is in the k-th
762
00:40:37,150 --> 00:40:40,750
interval, and 'delta x sub
k' is just 'x sub k'
763
00:40:40,750 --> 00:40:43,130
minus 'x sub 'k - 1''.
764
00:40:43,130 --> 00:40:48,200
If we take that limit as the
largest 'delta x' approaches 0
765
00:40:48,200 --> 00:40:52,960
and call that 'Q', that
limit 'Q' exists.
766
00:40:52,960 --> 00:40:55,940
Symbolically, it's written by
the definite integral from 'a'
767
00:40:55,940 --> 00:41:01,660
to 'b', ''f of x' dx', and,
more to the point, if you
768
00:41:01,660 --> 00:41:05,200
happen to know differential
calculus, you can compute 'Q'
769
00:41:05,200 --> 00:41:09,430
just by computing 'G of b'
minus 'G of a', where 'G
770
00:41:09,430 --> 00:41:13,530
prime' is any function whose
derivative is 'f'.
771
00:41:13,530 --> 00:41:14,290
OK?
772
00:41:14,290 --> 00:41:16,800
Now again, this is why I'm
calling it a summary.
773
00:41:16,800 --> 00:41:20,570
If you separate this out from
all of the computational stuff
774
00:41:20,570 --> 00:41:23,010
that we did in today's lecture,
this is the part
775
00:41:23,010 --> 00:41:24,680
that's left.
776
00:41:24,680 --> 00:41:25,500
OK?
777
00:41:25,500 --> 00:41:29,080
And what I want to do next
time is to show you that
778
00:41:29,080 --> 00:41:32,170
things are not quite this
straightforward all the time,
779
00:41:32,170 --> 00:41:34,860
that certain nice things have
been happening here that allow
780
00:41:34,860 --> 00:41:36,820
us, essentially, to get
away with murder.
781
00:41:36,820 --> 00:41:40,570
And what I mean by that will
become clearer next time, but
782
00:41:40,570 --> 00:41:42,220
until next time then goodbye.
783
00:41:42,220 --> 00:41:45,130
784
00:41:45,130 --> 00:41:47,510
MALE ANNOUNCER: Funding for the
publication of this video
785
00:41:47,510 --> 00:41:52,380
was provided by the Gabriella
and Paul Rosenbaum Foundation.
786
00:41:52,380 --> 00:41:56,560
Help OCW continue to provide
free and open access to MIT
787
00:41:56,560 --> 00:42:00,760
courses by making a donation
at ocw.mit.edu/donate.
788
00:42:00,760 --> 00:42:05,503