1 00:00:00,490 –> 00:00:02,150 Hello and welcome back

2 00:00:02,160 –> 00:00:04,070 to real analysis.

3 00:00:04,960 –> 00:00:06,510 And as always, first, I want

4 00:00:06,519 –> 00:00:08,000 to thank all the nice people

5 00:00:08,010 –> 00:00:09,020 that support this channel

6 00:00:09,029 –> 00:00:10,489 on study or paypal.

7 00:00:11,500 –> 00:00:12,880 Now, as you can see, we’ve

8 00:00:12,890 –> 00:00:14,399 reached part 48 in the

9 00:00:14,409 –> 00:00:15,260 series.

10 00:00:15,359 –> 00:00:17,010 And now we will start talking

11 00:00:17,020 –> 00:00:18,600 about the Riemann integral.

12 00:00:19,450 –> 00:00:20,909 Indeed, this will be the

13 00:00:20,920 –> 00:00:22,549 last chapter we will cover

14 00:00:22,559 –> 00:00:24,190 in this series about real

15 00:00:24,200 –> 00:00:24,950 analysis.

16 00:00:25,819 –> 00:00:27,350 You will see this notion

17 00:00:27,360 –> 00:00:28,920 of an integral is as

18 00:00:28,930 –> 00:00:30,850 important as the one we

19 00:00:30,860 –> 00:00:32,200 already introduced.

20 00:00:32,759 –> 00:00:34,479 Also this idea can be

21 00:00:34,490 –> 00:00:36,040 nicely visualized with the

22 00:00:36,049 –> 00:00:37,130 graph of a function.

23 00:00:38,229 –> 00:00:39,500 So here you see the graph

24 00:00:39,509 –> 00:00:41,189 of the function F which should

25 00:00:41,200 –> 00:00:43,060 be defined on the interval

26 00:00:43,069 –> 00:00:43,250 A

27 00:00:44,830 –> 00:00:45,330 OK.

28 00:00:45,340 –> 00:00:46,889 Now you might already know

29 00:00:47,139 –> 00:00:48,560 the integral of the function

30 00:00:48,569 –> 00:00:50,209 F is the orientated

31 00:00:50,220 –> 00:00:51,689 area between the

32 00:00:51,700 –> 00:00:53,630 graph and the X axis.

33 00:00:54,650 –> 00:00:56,430 We say orientated

34 00:00:56,490 –> 00:00:57,610 because there could be a

35 00:00:57,619 –> 00:00:58,830 sign involved

36 00:00:59,900 –> 00:01:00,880 more precisely.

37 00:01:00,889 –> 00:01:02,130 This means that the parts

38 00:01:02,139 –> 00:01:03,860 that are above the X axis

39 00:01:03,869 –> 00:01:05,830 are positive and the parts

40 00:01:05,839 –> 00:01:07,309 that are below the X axis

41 00:01:07,319 –> 00:01:08,319 are negative.

42 00:01:09,089 –> 00:01:10,459 Hence, at the moment for

43 00:01:10,470 –> 00:01:11,669 this picture, it does not

44 00:01:11,680 –> 00:01:12,989 matter because everything

45 00:01:13,000 –> 00:01:14,349 here is above the X axis.

46 00:01:14,360 –> 00:01:16,349 So we have a positive area.

47 00:01:17,279 –> 00:01:17,779 OK.

48 00:01:17,790 –> 00:01:19,160 But then the question is

49 00:01:19,169 –> 00:01:20,610 how can we calculate this

50 00:01:20,620 –> 00:01:21,059 area?

51 00:01:21,069 –> 00:01:23,040 Then now I can

52 00:01:23,050 –> 00:01:25,010 already tell you the overall

53 00:01:25,019 –> 00:01:26,860 idea is that we approximate

54 00:01:26,870 –> 00:01:28,230 it with rectangles.

55 00:01:28,970 –> 00:01:30,500 Indeed, the procedure for

56 00:01:30,510 –> 00:01:31,970 the Riemann integral is that

57 00:01:31,980 –> 00:01:33,830 one chooses points on the

58 00:01:33,839 –> 00:01:35,809 X axis and then draws

59 00:01:35,819 –> 00:01:37,510 rectangles above them.

60 00:01:38,949 –> 00:01:40,599 So you should see if we look

61 00:01:40,610 –> 00:01:42,449 at one rectangle here, we

62 00:01:42,459 –> 00:01:44,230 already have the width and

63 00:01:44,239 –> 00:01:45,449 the height is then given

64 00:01:45,459 –> 00:01:47,080 by one value of the function

65 00:01:47,089 –> 00:01:48,550 F inside this

66 00:01:48,559 –> 00:01:49,169 interval.

67 00:01:50,010 –> 00:01:51,360 For example, the picture

68 00:01:51,370 –> 00:01:52,720 then could look like this

69 00:01:52,730 –> 00:01:54,680 where we have six rectangles.

70 00:01:55,629 –> 00:01:57,510 And of course, the area of

71 00:01:57,519 –> 00:01:58,809 a rectangle can be

72 00:01:58,819 –> 00:02:00,440 calculated in a simple way.

73 00:02:01,269 –> 00:02:02,989 It’s just the width times

74 00:02:03,000 –> 00:02:03,569 the height.

75 00:02:04,540 –> 00:02:05,849 And then we just have to

76 00:02:05,860 –> 00:02:07,610 sum up all the areas of the

77 00:02:07,620 –> 00:02:08,970 rectangles to get an

78 00:02:08,979 –> 00:02:10,880 approximation of the area

79 00:02:10,889 –> 00:02:11,880 we are interested in.

80 00:02:12,800 –> 00:02:14,630 Hence, when we give the points

81 00:02:14,639 –> 00:02:16,059 here a name on the X axis.

82 00:02:16,070 –> 00:02:17,750 So maybe it’s simply X one

83 00:02:17,759 –> 00:02:18,979 X two and so on.

84 00:02:19,139 –> 00:02:20,740 Then the width of this

85 00:02:20,750 –> 00:02:22,479 rectangle here is simply

86 00:02:22,490 –> 00:02:24,380 X three minus X two.

87 00:02:25,139 –> 00:02:26,660 So in general, we can just

88 00:02:26,669 –> 00:02:28,660 write xj minus xj-1

89 00:02:28,669 –> 00:02:29,399 .

90 00:02:30,179 –> 00:02:31,690 And then in order to get

91 00:02:31,699 –> 00:02:33,279 the area of the rectangle,

92 00:02:33,330 –> 00:02:34,880 we have to multiply with

93 00:02:34,889 –> 00:02:35,539 the height

94 00:02:36,610 –> 00:02:38,419 for this the point, we choose

95 00:02:38,429 –> 00:02:40,139 an interval to get the value

96 00:02:40,149 –> 00:02:41,619 of the function we call

97 00:02:41,639 –> 00:02:42,339 cj.

98 00:02:43,389 –> 00:02:45,240 Therefore, here you see this

99 00:02:45,250 –> 00:02:46,839 is the area of one

100 00:02:46,850 –> 00:02:47,639 rectangle.

101 00:02:48,509 –> 00:02:50,139 So in the last step, we simply

102 00:02:50,149 –> 00:02:52,009 have the sum of all

103 00:02:52,020 –> 00:02:53,210 these areas.

104 00:02:54,160 –> 00:02:55,559 And in general, we can say

105 00:02:55,570 –> 00:02:57,169 we have N rectangles that

106 00:02:57,179 –> 00:02:58,259 are involved here.

107 00:02:59,279 –> 00:02:59,559 OK.

108 00:02:59,570 –> 00:03:01,149 Then soon we will see how

109 00:03:01,160 –> 00:03:02,839 we can do a limit process

110 00:03:02,850 –> 00:03:04,720 and to infinity to get the

111 00:03:04,729 –> 00:03:05,839 actual area.

112 00:03:06,740 –> 00:03:08,419 In fact, instead of the sum,

113 00:03:08,429 –> 00:03:09,880 we will use this long as

114 00:03:09,889 –> 00:03:10,979 as a symbol.

115 00:03:11,639 –> 00:03:13,279 Otherwise, it looks similarly

116 00:03:13,289 –> 00:03:14,839 because F is involved.

117 00:03:14,850 –> 00:03:16,429 And instead of this difference

118 00:03:16,440 –> 00:03:18,130 here, we write DX.

119 00:03:19,009 –> 00:03:20,410 So this is the symbol for

120 00:03:20,419 –> 00:03:21,740 the Riemann integral.

121 00:03:21,750 –> 00:03:23,300 And often we put the bounds

122 00:03:23,309 –> 00:03:24,750 of the interval A and

123 00:03:24,759 –> 00:03:26,229 B at the symbol.

124 00:03:27,259 –> 00:03:27,720 OK.

125 00:03:27,729 –> 00:03:29,160 What you really should remember

126 00:03:29,169 –> 00:03:30,470 here is that for the Riemann

127 00:03:30,479 –> 00:03:32,179 integral, we always start

128 00:03:32,190 –> 00:03:33,429 with a partition of the X

129 00:03:33,520 –> 00:03:33,979 axis.

130 00:03:34,899 –> 00:03:36,500 And then with some procedure

131 00:03:36,509 –> 00:03:38,279 I explain, soon we get to

132 00:03:38,289 –> 00:03:39,669 the definition of the Riemann

133 00:03:39,699 –> 00:03:41,639 integral of a function F

134 00:03:42,720 –> 00:03:44,419 I always specify the name

135 00:03:44,580 –> 00:03:46,080 Riemann here because there

136 00:03:46,089 –> 00:03:47,740 is another important notion

137 00:03:47,869 –> 00:03:49,669 a more modern one which is

138 00:03:49,679 –> 00:03:51,139 called the integral.

139 00:03:52,050 –> 00:03:53,789 Of course, in the end, both

140 00:03:53,800 –> 00:03:54,970 definitions will describe

141 00:03:54,979 –> 00:03:56,720 the same thing, namely the

142 00:03:56,729 –> 00:03:58,470 orientated area between the

143 00:03:58,479 –> 00:03:59,929 graph and the X axis.

144 00:04:00,610 –> 00:04:02,059 However, it turns out that

145 00:04:02,070 –> 00:04:03,429 the la back integral works

146 00:04:03,440 –> 00:04:05,089 more generally, which means

147 00:04:05,100 –> 00:04:06,910 you can apply to more functions.

148 00:04:07,720 –> 00:04:09,619 But since the Riemann is

149 00:04:09,630 –> 00:04:11,179 not hard to explain, it’s

150 00:04:11,190 –> 00:04:12,460 good to start with it.

151 00:04:13,699 –> 00:04:15,199 If you are interested in

152 00:04:15,210 –> 00:04:16,380 the Lebesgue integral, I have

153 00:04:16,390 –> 00:04:18,089 a whole other series about

154 00:04:18,130 –> 00:04:18,149 it.

155 00:04:18,988 –> 00:04:19,329 OK.

156 00:04:19,339 –> 00:04:21,079 However, here we will explain

157 00:04:21,089 –> 00:04:22,470 the Riemann integral which

158 00:04:22,480 –> 00:04:23,779 means we have to start with

159 00:04:23,790 –> 00:04:25,470 the partition of the X axis.

160 00:04:26,160 –> 00:04:27,989 More precisely, we will define

161 00:04:28,000 –> 00:04:29,299 the name partition of the

162 00:04:29,309 –> 00:04:31,000 compact interval A B.

163 00:04:31,730 –> 00:04:32,869 Now you have already seen

164 00:04:32,880 –> 00:04:34,709 above it’s simply a set that

165 00:04:34,720 –> 00:04:36,709 consists of points we choose

166 00:04:36,720 –> 00:04:38,029 inside the interval.

167 00:04:39,000 –> 00:04:40,649 Indeed, what one often does

168 00:04:40,660 –> 00:04:42,059 is to include the boundary

169 00:04:42,070 –> 00:04:44,000 points A and B as well.

170 00:04:44,899 –> 00:04:46,160 Therefore, we start at the

171 00:04:46,170 –> 00:04:48,070 point X zero, then comes

172 00:04:48,079 –> 00:04:49,940 X one X two and

173 00:04:49,950 –> 00:04:50,440 so on.

174 00:04:51,220 –> 00:04:52,579 And the last point should

175 00:04:52,589 –> 00:04:53,519 be XN.

176 00:04:54,450 –> 00:04:55,869 So you see it should be a

177 00:04:55,880 –> 00:04:57,049 finite set

178 00:04:58,040 –> 00:04:59,609 and the defining property

179 00:04:59,619 –> 00:05:01,230 of the set you also see

180 00:05:01,239 –> 00:05:01,769 above.

181 00:05:01,859 –> 00:05:03,399 So we have an order of the

182 00:05:03,410 –> 00:05:03,980 points.

183 00:05:04,929 –> 00:05:06,679 Hence X zero is the smallest

184 00:05:06,690 –> 00:05:08,339 one, then comes X one and

185 00:05:08,350 –> 00:05:08,839 so on.

186 00:05:08,880 –> 00:05:10,630 So we have strict inequalities

187 00:05:10,640 –> 00:05:12,230 here and

188 00:05:12,239 –> 00:05:13,950 also X zero should also

189 00:05:13,959 –> 00:05:15,679 be A and XN

190 00:05:15,690 –> 00:05:16,709 should be B

191 00:05:17,940 –> 00:05:18,480 OK.

192 00:05:18,489 –> 00:05:20,440 So this is the whole definition

193 00:05:20,450 –> 00:05:22,200 of the notion partition

194 00:05:22,209 –> 00:05:23,239 of an interval.

195 00:05:24,049 –> 00:05:25,750 Now for such a partition

196 00:05:25,790 –> 00:05:27,739 and a function F, we could

197 00:05:27,750 –> 00:05:29,470 calculate this sum here.

198 00:05:30,230 –> 00:05:31,910 However, because this could

199 00:05:31,920 –> 00:05:33,640 be hard, we should first

200 00:05:33,649 –> 00:05:35,329 start with functions F that

201 00:05:35,339 –> 00:05:36,809 are not so complicated.

202 00:05:37,630 –> 00:05:39,000 In fact, the functions we

203 00:05:39,010 –> 00:05:40,500 now consider are usually

204 00:05:40,510 –> 00:05:42,079 called step functions

205 00:05:43,070 –> 00:05:44,200 to make everything simpler

206 00:05:44,209 –> 00:05:44,890 to read.

207 00:05:44,899 –> 00:05:46,700 I now use Greek letters

208 00:05:46,709 –> 00:05:47,940 for the step functions.

209 00:05:48,920 –> 00:05:50,470 So here we have a lower case

210 00:05:50,480 –> 00:05:52,450 phi so the

211 00:05:52,459 –> 00:05:53,589 function should be defined

212 00:05:53,600 –> 00:05:55,570 on the interval A B and we

213 00:05:55,579 –> 00:05:57,380 call it a step function if

214 00:05:57,390 –> 00:05:59,130 it’s piece wisely constant.

215 00:05:59,880 –> 00:06:01,429 This means if we look at

216 00:06:01,440 –> 00:06:03,070 the graph, we only see

217 00:06:03,079 –> 00:06:04,339 horizontal lines.

218 00:06:05,279 –> 00:06:06,980 Therefore maybe let’s immediately

219 00:06:06,989 –> 00:06:08,049 visualize this.

220 00:06:09,190 –> 00:06:11,179 For example, maybe the function

221 00:06:11,190 –> 00:06:12,559 starts with one, will you

222 00:06:12,570 –> 00:06:14,239 hear then comes

223 00:06:14,250 –> 00:06:15,589 another value here

224 00:06:16,730 –> 00:06:18,230 and then the next one could

225 00:06:18,239 –> 00:06:18,709 be here.

226 00:06:18,720 –> 00:06:20,540 For example, so you

227 00:06:20,549 –> 00:06:21,899 see it doesn’t have to be

228 00:06:21,910 –> 00:06:23,579 a complete constant function.

229 00:06:23,589 –> 00:06:24,839 There could be jumps, there

230 00:06:24,850 –> 00:06:26,109 could be a lot of jumps but

231 00:06:26,119 –> 00:06:27,470 only finally many,

232 00:06:28,350 –> 00:06:29,869 which means for example,

233 00:06:29,880 –> 00:06:30,890 you could restrict it to

234 00:06:30,899 –> 00:06:32,380 an interval maybe here.

235 00:06:32,549 –> 00:06:33,869 And then you have a constant

236 00:06:33,880 –> 00:06:34,329 function,

237 00:06:35,489 –> 00:06:36,959 maybe it’s also important

238 00:06:36,970 –> 00:06:38,230 to note here that the

239 00:06:38,239 –> 00:06:40,119 values at the jump points

240 00:06:40,130 –> 00:06:41,380 don’t matter at all.

241 00:06:42,290 –> 00:06:44,109 For example, this is allowed

242 00:06:44,119 –> 00:06:44,709 to happen.

243 00:06:45,929 –> 00:06:47,339 And maybe you already know

244 00:06:47,350 –> 00:06:49,230 why this is because the

245 00:06:49,239 –> 00:06:51,209 area below this curve

246 00:06:51,220 –> 00:06:53,149 to the x axis does not

247 00:06:53,160 –> 00:06:54,820 care what this will is.

248 00:06:55,809 –> 00:06:57,350 And of course, in the end,

249 00:06:57,359 –> 00:06:58,470 we will be interested in

250 00:06:58,480 –> 00:07:00,290 this area because it’s the

251 00:07:00,299 –> 00:07:01,070 integral.

252 00:07:02,070 –> 00:07:02,570 OK.

253 00:07:02,579 –> 00:07:03,850 Then let’s describe this

254 00:07:03,859 –> 00:07:05,250 definition of a piece wise

255 00:07:05,260 –> 00:07:07,109 constant function in a formal

256 00:07:07,119 –> 00:07:07,410 way.

257 00:07:08,399 –> 00:07:09,820 And as you might guess we

258 00:07:09,829 –> 00:07:11,619 can use a partition here for

259 00:07:11,630 –> 00:07:12,899 the interval A B,

260 00:07:13,809 –> 00:07:15,149 this means that the best

261 00:07:15,160 –> 00:07:16,869 idea would be to set these

262 00:07:16,880 –> 00:07:18,429 points X one X two and so

263 00:07:18,440 –> 00:07:20,019 on at these jump

264 00:07:20,029 –> 00:07:21,839 points because

265 00:07:21,850 –> 00:07:23,500 then we have the corresponding

266 00:07:23,510 –> 00:07:25,250 intervals where the function

267 00:07:25,260 –> 00:07:26,239 is constant.

268 00:07:27,059 –> 00:07:28,809 However, then it’s also no

269 00:07:28,820 –> 00:07:30,769 problem at all fixing these

270 00:07:30,779 –> 00:07:32,000 constant values.

271 00:07:32,869 –> 00:07:34,140 So let’s call them C one

272 00:07:34,149 –> 00:07:35,690 C two and so on.

273 00:07:36,709 –> 00:07:38,100 Indeed, what could happen

274 00:07:38,109 –> 00:07:39,420 is that some values

275 00:07:39,429 –> 00:07:40,250 coincide.

276 00:07:41,089 –> 00:07:42,079 So here we have that the

277 00:07:42,089 –> 00:07:43,779 value in this interval is

278 00:07:43,790 –> 00:07:45,470 the same as in this one.

279 00:07:46,339 –> 00:07:47,519 This is not a problem.

280 00:07:47,570 –> 00:07:48,679 The important thing is that

281 00:07:48,690 –> 00:07:50,339 for each interval, we have

282 00:07:50,350 –> 00:07:51,709 such a number CJ,

283 00:07:52,600 –> 00:07:54,320 which means we have any of

284 00:07:54,329 –> 00:07:54,670 them.

285 00:07:55,790 –> 00:07:56,290 OK.

286 00:07:56,299 –> 00:07:57,839 And now we can write down

287 00:07:57,850 –> 00:07:59,649 the defining property we’ve

288 00:07:59,660 –> 00:08:00,750 already discussed,

289 00:08:01,369 –> 00:08:03,299 meaning that we can restrict

290 00:08:03,309 –> 00:08:04,929 the function phi to an

291 00:08:04,940 –> 00:08:05,649 interval.

292 00:08:06,450 –> 00:08:08,019 Namely, we can do this for

293 00:08:08,029 –> 00:08:09,420 each interval that starts

294 00:08:09,429 –> 00:08:11,339 with XJ minus one and

295 00:08:11,350 –> 00:08:12,480 ends with XJ.

296 00:08:13,429 –> 00:08:14,890 Indeed, in order to avoid

297 00:08:14,899 –> 00:08:16,690 these jump points here, we

298 00:08:16,700 –> 00:08:18,250 need the open intervals.

299 00:08:19,109 –> 00:08:19,549 OK.

300 00:08:19,559 –> 00:08:20,970 And now we know restricted

301 00:08:20,980 –> 00:08:22,529 to this interval, it’s a

302 00:08:22,540 –> 00:08:23,690 constant function

303 00:08:24,559 –> 00:08:26,029 and the value is given by

304 00:08:26,040 –> 00:08:26,950 CJ.

305 00:08:27,880 –> 00:08:28,429 OK.

306 00:08:28,440 –> 00:08:29,820 And of course, every time

307 00:08:29,829 –> 00:08:31,089 you see something like this,

308 00:08:31,119 –> 00:08:32,729 it means it should hold for

309 00:08:32,739 –> 00:08:34,650 all possible values of J.

310 00:08:35,650 –> 00:08:37,308 Well, now you know what we

311 00:08:37,320 –> 00:08:38,909 call a step function when

312 00:08:38,919 –> 00:08:40,390 we want to define the Riemann

313 00:08:40,400 –> 00:08:41,070 integral.

314 00:08:42,109 –> 00:08:43,629 So you see it’s not

315 00:08:43,638 –> 00:08:44,929 complicated at all.

316 00:08:45,780 –> 00:08:47,169 Also the integral should

317 00:08:47,179 –> 00:08:48,849 not be complicated because

318 00:08:48,859 –> 00:08:50,729 we immediately see the rectangles

319 00:08:50,739 –> 00:08:51,880 here we could use

320 00:08:52,539 –> 00:08:54,200 and then these rectangles

321 00:08:54,210 –> 00:08:55,929 exactly describe the

322 00:08:55,940 –> 00:08:57,799 area between the graph and

323 00:08:57,809 –> 00:08:58,739 the X axis.

324 00:08:59,580 –> 00:09:00,820 Therefore, my question for

325 00:09:00,830 –> 00:09:02,520 you would be, can we immediately

326 00:09:02,530 –> 00:09:04,229 define the integral for

327 00:09:04,239 –> 00:09:05,390 such a step function?

328 00:09:06,219 –> 00:09:07,820 Please recall this is the

329 00:09:07,830 –> 00:09:09,640 symbol that is a number

330 00:09:09,650 –> 00:09:11,559 that represents the orientated

331 00:09:11,570 –> 00:09:12,200 area.

332 00:09:13,080 –> 00:09:14,770 This means in this case,

333 00:09:14,840 –> 00:09:16,369 we just have to add

334 00:09:16,380 –> 00:09:17,770 all the areas of the

335 00:09:17,780 –> 00:09:18,729 rectangles here.

336 00:09:19,559 –> 00:09:21,140 So we have a sum that starts

337 00:09:21,150 –> 00:09:23,020 with one and goes to N

338 00:09:23,799 –> 00:09:25,640 also we know the height is

339 00:09:25,650 –> 00:09:27,580 given by CJ and

340 00:09:27,590 –> 00:09:28,900 the width is given by the

341 00:09:28,909 –> 00:09:30,549 difference of these two points

342 00:09:30,559 –> 00:09:30,849 here.

343 00:09:31,820 –> 00:09:33,109 So what we see here is in

344 00:09:33,119 –> 00:09:35,049 this case, the integral

345 00:09:35,059 –> 00:09:36,690 is given by this finite

346 00:09:36,700 –> 00:09:37,200 sum.

347 00:09:38,049 –> 00:09:39,400 However, what we need to

348 00:09:39,409 –> 00:09:41,299 answer here is, is this a

349 00:09:41,309 –> 00:09:42,479 correct definition?

350 00:09:43,429 –> 00:09:45,030 Because at first, it seems

351 00:09:45,039 –> 00:09:46,940 that there are a lot of possibilities

352 00:09:46,950 –> 00:09:48,080 to choose the petition of

353 00:09:48,090 –> 00:09:49,960 the interval and also to

354 00:09:49,969 –> 00:09:51,809 choose the numbers CJ.

355 00:09:52,669 –> 00:09:54,010 However, here on the left

356 00:09:54,020 –> 00:09:55,440 hand side, neither the

357 00:09:55,450 –> 00:09:57,210 partition nor the number

358 00:09:57,219 –> 00:09:58,409 CJ occur.

359 00:09:59,500 –> 00:10:00,140 In summary.

360 00:10:00,150 –> 00:10:01,659 What we need to show is that

361 00:10:01,669 –> 00:10:03,309 this here is well

362 00:10:03,320 –> 00:10:03,979 defined.

363 00:10:04,909 –> 00:10:06,539 OK, then please think about

364 00:10:06,549 –> 00:10:08,049 it and then we can talk about

365 00:10:08,059 –> 00:10:09,809 it together in the next video.

366 00:10:10,679 –> 00:10:11,960 Therefore, I hope I see you

367 00:10:11,969 –> 00:10:13,429 there and have a nice day.

368 00:10:13,549 –> 00:10:14,280 Bye.

Q1: What is the Riemann integral of the function $f: [0,1] \rightarrow \mathbb{R}$ with $f(x) = 1$?

A1: 0

A2: -1

A3: 2

A4: 1

Q2: Consider the intervall $[0,9]$. What is a correct partition of this interval?

A1: $ { 1,2,3 } $

A2: $ { 0,1,2,3 } $

A3: $ { 0,1,2,3,9 } $

A4: $ { 0,1,2,3,9,10 } $

A5: $ {-1,0,1,2,3,9,10 } $

A6: $ { -1,0,1,2,3,9,10,11 } $

Q3: Is the function $\phi : [0,1] \rightarrow \mathbb{R}$ with $$ \phi(x) = \begin{cases} 1 , , \text{ for } x = 0 \ 2 , , \text{ for } x \neq 0 \end{cases} $$ a step function?

A1: Yes!

A2: No!

Q4: Is the function $\phi : [0,1] \rightarrow \mathbb{R}$ with $ \phi(x) = x$ a step function?

A1: Yes!

A2: No!