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Title: Cavalieri’s principle - An example
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Series: Measure Theory
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YouTube-Title: Measure Theory 18 | Cavalieri’s principle - An example
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Subtitle on GitHub: mt18_sub_eng.srt
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Subtitle in English
1 00:00:00,649 –> 00:00:02,480 Hello and welcome back to
2 00:00:02,490 –> 00:00:04,190 a new video about measure
3 00:00:04,199 –> 00:00:04,820 theory.
4 00:00:05,679 –> 00:00:07,099 And first, I want to thank
5 00:00:07,110 –> 00:00:08,939 all the nice supporters on
6 00:00:08,949 –> 00:00:09,449 Steady
7 00:00:10,399 –> 00:00:12,140 extending the last video.
8 00:00:12,149 –> 00:00:13,979 I want to show you here an
9 00:00:13,989 –> 00:00:15,979 example of Cavalieri’s
10 00:00:15,989 –> 00:00:16,780 principle.
11 00:00:17,489 –> 00:00:19,159 This should be a comprehensible
12 00:00:19,170 –> 00:00:19,979 example.
13 00:00:19,989 –> 00:00:21,750 And therefore we focus on
14 00:00:21,760 –> 00:00:23,649 the normal volume in
15 00:00:23,659 –> 00:00:24,350 R three.
16 00:00:25,379 –> 00:00:26,899 The typical example can look
17 00:00:26,909 –> 00:00:28,370 like this, calculate the
18 00:00:28,379 –> 00:00:30,360 volume of the pyramid with
19 00:00:30,370 –> 00:00:31,879 the following corners.
20 00:00:33,090 –> 00:00:34,400 The word volume here
21 00:00:34,409 –> 00:00:36,349 means the Lebesgue measure
22 00:00:36,360 –> 00:00:37,669 on R³.
23 00:00:38,700 –> 00:00:39,880 And what you also learned
24 00:00:39,889 –> 00:00:41,639 in the last video is that
25 00:00:41,650 –> 00:00:43,150 you get this Lebesgue measure
26 00:00:43,159 –> 00:00:44,540 in R three by a
27 00:00:44,549 –> 00:00:46,430 product measure construction
28 00:00:46,569 –> 00:00:48,439 starting with the Lebesgue measure
29 00:00:48,450 –> 00:00:49,139 in R.
30 00:00:50,049 –> 00:00:51,630 However, before we talk about
31 00:00:51,639 –> 00:00:53,080 this, let’s start by
32 00:00:53,090 –> 00:00:54,830 visualizing the pyramid with
33 00:00:54,840 –> 00:00:55,869 a short sketch
34 00:00:57,619 –> 00:00:57,889 Here,
35 00:00:57,900 –> 00:00:59,360 it’s not hard to see that
36 00:00:59,369 –> 00:01:00,209 exactly this point
37 00:01:00,220 –> 00:01:02,099 (0,0,1) is
38 00:01:02,110 –> 00:01:04,089 the top of the pyramid and
39 00:01:04,099 –> 00:01:05,760 all the other points
40 00:01:05,769 –> 00:01:07,580 form the base, the
41 00:01:07,589 –> 00:01:09,040 square of the pyramid.
42 00:01:10,239 –> 00:01:11,610 In a formal way, the pyramid
43 00:01:11,620 –> 00:01:13,129 is given as a subset.
44 00:01:13,139 –> 00:01:14,660 And let’s call it K
45 00:01:15,080 –> 00:01:16,419 in R three.
46 00:01:18,790 –> 00:01:20,500 By looking at our sketch,
47 00:01:20,540 –> 00:01:22,010 we are able to write
48 00:01:22,019 –> 00:01:23,980 down a formal definition
49 00:01:23,989 –> 00:01:25,250 of this set K.
50 00:01:26,349 –> 00:01:28,250 It’s the set of all the points
51 00:01:28,260 –> 00:01:29,510 xyz in
52 00:01:30,129 –> 00:01:31,330 R three with the
53 00:01:31,339 –> 00:01:33,129 following conditions.
54 00:01:34,669 –> 00:01:36,459 The first thing we see immediately
55 00:01:36,470 –> 00:01:38,110 is that Z lies
56 00:01:38,120 –> 00:01:39,410 between zero and one.
57 00:01:39,419 –> 00:01:40,309 So we can write
58 00:01:40,860 –> 00:01:42,660 zero less or equal
59 00:01:42,669 –> 00:01:44,580 than Z less or equal
60 00:01:44,589 –> 00:01:45,209 than one.
61 00:01:46,620 –> 00:01:47,800 You can also see that the
62 00:01:47,809 –> 00:01:49,500 possible values for X and
63 00:01:49,510 –> 00:01:51,199 Y depend
64 00:01:51,209 –> 00:01:52,480 on this X.
65 00:01:53,400 –> 00:01:55,019 Now, if you think about it,
66 00:01:55,029 –> 00:01:56,900 which functions describe
67 00:01:56,910 –> 00:01:58,819 these planes here, then you
68 00:01:58,830 –> 00:02:00,300 find that the short
69 00:02:00,309 –> 00:02:02,059 formulation is given by the
70 00:02:02,069 –> 00:02:03,779 absolute value is less or
71 00:02:03,790 –> 00:02:05,599 equal than one minus set.
72 00:02:05,760 –> 00:02:07,480 And the same for
73 00:02:07,489 –> 00:02:08,020 Y
74 00:02:10,850 –> 00:02:11,190 OK.
75 00:02:11,199 –> 00:02:12,630 I think here also another
76 00:02:12,639 –> 00:02:14,470 visualization is appropriate.
77 00:02:14,600 –> 00:02:16,190 Let’s look at the pyramid
78 00:02:16,199 –> 00:02:17,270 from the top.
79 00:02:18,089 –> 00:02:19,479 Then we can see what happens
80 00:02:19,490 –> 00:02:21,089 to the X and Y values.
81 00:02:21,100 –> 00:02:23,039 And for the Z value, we use
82 00:02:23,050 –> 00:02:24,160 different colours.
83 00:02:24,880 –> 00:02:26,339 For example, here we have
84 00:02:26,350 –> 00:02:27,850 the square of the bottom
85 00:02:28,429 –> 00:02:30,039 which corresponds to the
86 00:02:30,050 –> 00:02:31,910 Z value equals to
87 00:02:31,919 –> 00:02:32,470 zero.
88 00:02:33,479 –> 00:02:35,050 This is what we call a level
89 00:02:35,059 –> 00:02:35,669 curve.
90 00:02:35,809 –> 00:02:37,250 However, if you look at an
91 00:02:37,259 –> 00:02:39,100 actual mountain and a map,
92 00:02:39,110 –> 00:02:40,509 you would call it a contour
93 00:02:40,520 –> 00:02:40,919 line.
94 00:02:41,899 –> 00:02:43,800 Now we can do this for different
95 00:02:43,809 –> 00:02:44,320 levels.
96 00:02:44,330 –> 00:02:45,690 For example, z
97 00:02:45,699 –> 00:02:46,399 equals
98 00:02:46,410 –> 00:02:48,100 1/4 would be
99 00:02:48,110 –> 00:02:50,000 this line here or this
100 00:02:50,009 –> 00:02:51,779 square and Z
101 00:02:51,789 –> 00:02:53,570 equals to one half
102 00:02:53,580 –> 00:02:55,080 would be this square.
103 00:02:55,820 –> 00:02:57,110 And in this picture at the
104 00:02:57,119 –> 00:02:58,800 top of the pyramid is
105 00:02:58,809 –> 00:03:00,479 just this point which
106 00:03:00,490 –> 00:03:02,320 corresponds to Z equals
107 00:03:02,330 –> 00:03:03,990 one, this
108 00:03:04,000 –> 00:03:05,429 nice picture now
109 00:03:05,440 –> 00:03:07,229 shows us what to do
110 00:03:08,009 –> 00:03:09,630 because we know how to
111 00:03:09,639 –> 00:03:11,309 calculate the measure of
112 00:03:11,320 –> 00:03:13,029 these squares here, which
113 00:03:13,039 –> 00:03:14,460 means the area of the
114 00:03:14,470 –> 00:03:15,190 squares.
115 00:03:15,929 –> 00:03:17,270 And then Cavalieri’s
116 00:03:17,279 –> 00:03:19,229 principle tells us that
117 00:03:19,240 –> 00:03:20,750 we have to integrate
118 00:03:20,759 –> 00:03:21,710 all these
119 00:03:21,720 –> 00:03:23,559 measures to get
120 00:03:23,570 –> 00:03:25,330 the volume of the permit
121 00:03:25,339 –> 00:03:25,660 here.
122 00:03:26,419 –> 00:03:27,889 Or in other words, if we
123 00:03:27,899 –> 00:03:29,789 call our Lebesgue measure
124 00:03:29,800 –> 00:03:31,720 in our three mu,
125 00:03:32,839 –> 00:03:34,520 then we can use our product
126 00:03:34,529 –> 00:03:36,259 measure construction to
127 00:03:36,270 –> 00:03:38,100 write it as the product measure
128 00:03:38,110 –> 00:03:40,000 of MU one and mu
129 00:03:40,009 –> 00:03:40,380 two.
130 00:03:41,380 –> 00:03:43,369 Here mu one is the,
131 00:03:43,380 –> 00:03:45,110 the Lebesgue measure on R
132 00:03:45,369 –> 00:03:46,740 which means the z
133 00:03:46,750 –> 00:03:47,649 coordinate.
134 00:03:48,509 –> 00:03:50,320 And mu two is still a back
135 00:03:50,330 –> 00:03:52,289 measure on R two, which
136 00:03:52,300 –> 00:03:54,169 means the X and Y
137 00:03:54,179 –> 00:03:55,570 coordinate together.
138 00:03:56,630 –> 00:03:58,279 At this point, we can use
139 00:03:58,289 –> 00:03:59,789 Cavalieri’s principle
140 00:04:00,320 –> 00:04:01,910 which tells us that the
141 00:04:01,919 –> 00:04:03,750 volume of the pyramid, which
142 00:04:03,759 –> 00:04:05,399 means mu of
143 00:04:05,410 –> 00:04:06,860 K is
144 00:04:06,869 –> 00:04:07,949 equal to the
145 00:04:07,960 –> 00:04:09,490 integral over
146 00:04:09,500 –> 00:04:10,229 R.
147 00:04:11,229 –> 00:04:12,649 This is the space where Mu
148 00:04:12,660 –> 00:04:13,669 one lives on.
149 00:04:13,679 –> 00:04:15,169 So we will integrate with
150 00:04:15,179 –> 00:04:16,450 respect to Mu one
151 00:04:17,338 –> 00:04:19,118 and we integrate the measures
152 00:04:19,128 –> 00:04:21,079 of these squares, which
153 00:04:21,088 –> 00:04:22,218 means mu
154 00:04:22,229 –> 00:04:23,928 two of the
155 00:04:23,938 –> 00:04:25,259 squares and we call them
156 00:04:25,269 –> 00:04:26,889 M with index Z
157 00:04:26,919 –> 00:04:27,799 zero.
158 00:04:28,368 –> 00:04:29,678 And therefore we integrate
159 00:04:29,688 –> 00:04:31,229 over the variable Z
160 00:04:31,278 –> 00:04:31,908 zero.
161 00:04:32,959 –> 00:04:34,299 Now, if you go back to the
162 00:04:34,309 –> 00:04:36,160 last video where I explained
163 00:04:36,170 –> 00:04:37,970 Cavalieri’s principle, you
164 00:04:37,980 –> 00:04:39,790 see that MZ
165 00:04:39,799 –> 00:04:41,670 zero are sets
166 00:04:41,679 –> 00:04:43,260 that are defined in the
167 00:04:43,269 –> 00:04:45,260 space where mu two lives.
168 00:04:45,269 –> 00:04:47,220 In this means
169 00:04:47,230 –> 00:04:49,070 in this case, we only have
170 00:04:49,079 –> 00:04:50,549 now points with two
171 00:04:50,559 –> 00:04:52,500 variables X and
172 00:04:52,510 –> 00:04:54,309 Y in R two.
173 00:04:55,350 –> 00:04:57,140 These are the sections through
174 00:04:57,149 –> 00:04:58,730 the pyramid where z is
175 00:04:58,739 –> 00:05:00,559 fixed as z zero.
176 00:05:00,760 –> 00:05:02,730 Therefore, we only have these
177 00:05:02,739 –> 00:05:04,529 two conditions here with
178 00:05:04,540 –> 00:05:05,929 z zero on the right.
179 00:05:07,130 –> 00:05:09,070 And this defines, then
180 00:05:09,350 –> 00:05:10,869 our MZ zero.
181 00:05:12,220 –> 00:05:13,799 Now, you should recognize
182 00:05:13,809 –> 00:05:15,440 these are exactly the
183 00:05:15,450 –> 00:05:17,279 squares from before.
184 00:05:19,730 –> 00:05:21,380 Hence, it’s no problem for
185 00:05:21,390 –> 00:05:23,350 us to calculate the area
186 00:05:23,359 –> 00:05:24,559 of such a square,
187 00:05:25,339 –> 00:05:27,170 the length of one side is
188 00:05:27,179 –> 00:05:29,160 given by two times
189 00:05:29,170 –> 00:05:30,720 one minus z zero.
190 00:05:31,359 –> 00:05:32,829 This is exactly the length
191 00:05:32,839 –> 00:05:34,339 we have in the X direction.
192 00:05:34,480 –> 00:05:36,239 And of course, it’s the same
193 00:05:36,250 –> 00:05:37,480 in the Y direction.
194 00:05:37,489 –> 00:05:39,309 Therefore, we can write this
195 00:05:39,399 –> 00:05:40,339 as a square.
196 00:05:41,440 –> 00:05:43,290 Obviously, this only makes
197 00:05:43,299 –> 00:05:45,079 sense if our z
198 00:05:45,089 –> 00:05:47,010 zero is chosen
199 00:05:47,019 –> 00:05:48,709 between zero and one.
200 00:05:48,720 –> 00:05:50,250 So it lies in interval
201 00:05:50,350 –> 00:05:52,339 0 to 1.
202 00:05:53,290 –> 00:05:54,730 Of course, this is given
203 00:05:54,739 –> 00:05:55,890 in the definition of the
204 00:05:55,899 –> 00:05:56,470 pyramid.
205 00:05:56,510 –> 00:05:58,230 A point lies only in the
206 00:05:58,239 –> 00:05:59,989 pyramid if the Z value
207 00:06:00,000 –> 00:06:01,790 lies between zero and one.
208 00:06:02,359 –> 00:06:03,790 And this is also how we should
209 00:06:03,799 –> 00:06:05,359 read this equality
210 00:06:05,369 –> 00:06:05,940 here.
211 00:06:05,950 –> 00:06:07,660 It only makes sense for Z
212 00:06:07,670 –> 00:06:08,850 between zero and one.
213 00:06:09,000 –> 00:06:10,820 Otherwise we should define
214 00:06:10,829 –> 00:06:12,779 M zero Z zero as
215 00:06:12,790 –> 00:06:13,549 the empty set.
216 00:06:14,489 –> 00:06:14,869 OK.
217 00:06:14,880 –> 00:06:16,350 So no confusion here.
218 00:06:16,359 –> 00:06:17,989 What you usually would do
219 00:06:18,040 –> 00:06:19,630 is just write down
220 00:06:20,739 –> 00:06:22,690 the integral restricted to
221 00:06:22,700 –> 00:06:23,670 the interval
222 00:06:23,679 –> 00:06:24,760 01.
223 00:06:25,700 –> 00:06:26,779 And then we can copy the
224 00:06:26,790 –> 00:06:28,440 function here which is
225 00:06:28,450 –> 00:06:30,100 four times one
226 00:06:30,109 –> 00:06:31,839 minus Z zero
227 00:06:31,890 –> 00:06:32,720 squared
228 00:06:33,459 –> 00:06:34,920 and here d mu 1
229 00:06:35,410 –> 00:06:36,839 Z zero.
230 00:06:37,709 –> 00:06:39,040 Because this is just a
231 00:06:39,049 –> 00:06:40,799 normal one dimensional
232 00:06:40,809 –> 00:06:42,410 integral with respect to
233 00:06:42,420 –> 00:06:44,160 the Lebesgue measure, we can
234 00:06:44,170 –> 00:06:45,730 use a shorter notation.
235 00:06:46,459 –> 00:06:47,859 We write the integral from
236 00:06:47,869 –> 00:06:49,570 0 to 1 copy
237 00:06:49,579 –> 00:06:50,859 again the function.
238 00:06:52,010 –> 00:06:53,709 And then we omit the measure
239 00:06:53,720 –> 00:06:55,640 D mu one and just write
240 00:06:55,649 –> 00:06:57,350 DZ zero
241 00:06:58,450 –> 00:06:59,890 to calculate this integral.
242 00:06:59,899 –> 00:07:01,869 You just need an antiderivative,
243 00:07:01,880 –> 00:07:03,309 which means here four
244 00:07:03,320 –> 00:07:04,269 times.
245 00:07:04,390 –> 00:07:05,470 And here we have
246 00:07:05,489 –> 00:07:06,429 minus
247 00:07:06,440 –> 00:07:08,070 1/3
248 00:07:08,510 –> 00:07:09,769 and one minus
249 00:07:09,779 –> 00:07:11,359 zero to the power of
250 00:07:11,369 –> 00:07:11,910 three.
251 00:07:12,519 –> 00:07:14,260 And the limits are
252 00:07:14,269 –> 00:07:15,820 zero and one,
253 00:07:17,209 –> 00:07:18,619 the upper part gives you
254 00:07:18,630 –> 00:07:19,350 zero.
255 00:07:19,440 –> 00:07:21,160 And the lower part here zero
256 00:07:21,170 –> 00:07:22,750 here gives you
257 00:07:22,760 –> 00:07:24,390 minus and minus.
258 00:07:24,399 –> 00:07:25,470 So plus
259 00:07:25,480 –> 00:07:27,380 4/3.
260 00:07:28,750 –> 00:07:30,279 And that’s how you apply
261 00:07:30,290 –> 00:07:31,730 Cavalieri’s principle.
262 00:07:31,739 –> 00:07:33,570 To calculate the volume
263 00:07:33,579 –> 00:07:34,549 of some body,
264 00:07:36,220 –> 00:07:37,829 you just look at it and then
265 00:07:37,839 –> 00:07:39,649 you think about how to
266 00:07:39,660 –> 00:07:41,440 section it such that you
267 00:07:41,450 –> 00:07:43,209 get out an integral that
268 00:07:43,220 –> 00:07:45,010 you can solve a
269 00:07:45,019 –> 00:07:46,850 good exercise would be now
270 00:07:46,859 –> 00:07:48,690 to do exactly the same
271 00:07:48,700 –> 00:07:50,010 thing and to
272 00:07:50,019 –> 00:07:51,809 calculate the volume of
273 00:07:51,820 –> 00:07:53,730 the sphere in R³.
274 00:07:54,600 –> 00:07:56,459 So please try this and
275 00:07:56,470 –> 00:07:58,410 after that, you are prepared
276 00:07:58,420 –> 00:08:00,109 to use Cavalieri’s principle
277 00:08:00,119 –> 00:08:01,440 also in other
278 00:08:01,450 –> 00:08:02,220 situations.
279 00:08:03,220 –> 00:08:04,869 Well then thanks for
280 00:08:04,880 –> 00:08:06,709 listening and see you next
281 00:08:06,720 –> 00:08:07,000 time.
282 00:08:07,660 –> 00:08:08,309 Bye.
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Last update: 2024-10
