1 00:00:00,000 --> 00:00:06,762 [music] There are some properties of the integral that are worth summarizing. 2 00:00:06,762 --> 00:00:12,740 For example, here's a property of integration that's often useful. 3 00:00:12,740 --> 00:00:17,870 If I want to integrate from a to b, right? That's what this last integral is. 4 00:00:17,870 --> 00:00:20,770 And it's the area under the graph between a and b. 5 00:00:20,770 --> 00:00:23,524 It's this whole region here that I've colored in. 6 00:00:23,525 --> 00:00:26,676 But I could split that up into two separate integrals, right? 7 00:00:26,676 --> 00:00:31,459 I could integrate from a to c at this first red integral, and then I could 8 00:00:31,459 --> 00:00:35,202 integrate from c to b, that's the second blue interval. 9 00:00:35,202 --> 00:00:40,281 And if I add together those two areas, I get the total area from a to b, right? 10 00:00:40,281 --> 00:00:46,100 And that geometric fact is exactly what's written down here, with symbols using our 11 00:00:46,100 --> 00:00:50,040 fancy integration notation. There's also a constant multiple rule. 12 00:00:50,040 --> 00:00:54,705 Well, here's the constant multiple rule. For some constant K, the integral from a 13 00:00:54,705 --> 00:00:59,730 to b of that constant times a function is that constant times the integral of that 14 00:00:59,730 --> 00:01:02,580 function. And this also makes sense geometrically. 15 00:01:02,580 --> 00:01:05,760 Here's two pictures. Here's the graph of y equals f of x. 16 00:01:05,760 --> 00:01:09,129 Here's the graph of y equals K times f of x. 17 00:01:09,130 --> 00:01:14,937 This thing here calculates this area, the area under the graph of K times f of x. 18 00:01:14,938 --> 00:01:19,170 And it's K times just the area under the graph of f of x. 19 00:01:19,170 --> 00:01:22,754 And it makes sense. Because if you take this graph and your 20 00:01:22,754 --> 00:01:27,027 stretch it K times, that multiplies the area by a factor of K. 21 00:01:27,028 --> 00:01:32,266 What about the intergral of a sum? Well, the integral of f of x plus g of x 22 00:01:32,266 --> 00:01:35,997 from a to b, right? That's this total area here. 23 00:01:35,998 --> 00:01:40,735 It's related to the area under the graph of f and the area under the graph of g, 24 00:01:40,735 --> 00:01:43,307 right? It's related to these integrals. 25 00:01:43,308 --> 00:01:50,380 Here, in green, I've sort of demonstrated what the area under the graph of f of x 26 00:01:50,380 --> 00:01:56,554 looks like with a specific Riemann sum. And in here, in red, I've drawn some 27 00:01:56,554 --> 00:02:01,852 rectangles for the Riemann sum of g. But they're, you know, sort of shifted up 28 00:02:01,852 --> 00:02:05,424 a bit. Because this curve here is the graph of y 29 00:02:05,424 --> 00:02:11,290 equals f of Xx plus g of x. So, the heights of these rectangles are 30 00:02:11,290 --> 00:02:16,705 actually what I would get if I were to just integrate g of x, right? 31 00:02:16,705 --> 00:02:23,326 The distance between f of x plus g of x, and f of x is exactly g of x here in red. 32 00:02:23,326 --> 00:02:28,920 So, this is kind of a proof by stacking, if you like, that the integral of f plus g 33 00:02:28,920 --> 00:02:31,930 is the integral of f plus the integral of g. 34 00:02:31,930 --> 00:02:36,553 A lot of these rules have analogs for the sigma notation stuff. 35 00:02:36,554 --> 00:02:40,928 We had this rule that said I could have pasted together integrals. 36 00:02:40,928 --> 00:02:46,136 And there's a corresponding rule for sum that says, if I sum f of the numbers 37 00:02:46,136 --> 00:02:51,680 between 1 and m, and then f of the numbers between m plus 1 and K, that's the same as 38 00:02:51,680 --> 00:02:56,575 applying f to all the numbers between 1 and K and adding that up, right? 39 00:02:56,575 --> 00:03:00,429 So, this same kinds of rules, I mean, there's an analogy there. 40 00:03:00,430 --> 00:03:04,738 Same kind of game here, right? I've got this constant multiple rule for 41 00:03:04,738 --> 00:03:09,180 integrals and I've got a corresponding constant multiple rule for sums. 42 00:03:09,180 --> 00:03:12,261 Of course, this constant multiple rule is just called distributivity, right? 43 00:03:12,261 --> 00:03:17,360 If I add up K times something, that's K times the sum of these things. 44 00:03:17,360 --> 00:03:22,638 But, it's the same kind of rule, right? I had this formula that said the integral 45 00:03:22,638 --> 00:03:27,129 of the sum is the sum of the integrals. We've got the same kind of formula for a 46 00:03:27,129 --> 00:03:29,816 sum, right? If I take a sum of f of n plus g of n, 47 00:03:29,816 --> 00:03:34,130 that's the same as adding up f of n for all the numbers between a and b. 48 00:03:34,130 --> 00:03:37,845 And then, adding to that g of n for all the numbers between a and b. 49 00:03:37,846 --> 00:03:42,000 And we're also seeing some similarities to the rules for derivatives. 50 00:03:42,000 --> 00:03:45,001 I've got the constant multiple rule for integrals. 51 00:03:45,002 --> 00:03:49,081 I've got a constant multiple rule for sums and I've got a constant multiple rule for 52 00:03:49,081 --> 00:03:51,833 derivatives. The derivative of a constant times some 53 00:03:51,833 --> 00:03:55,099 functions that constant times the derivative of the function. 54 00:03:56,130 --> 00:04:00,654 Same kind of deals for sums, right? I've got this sum of the integrals is the 55 00:04:00,654 --> 00:04:05,350 integral of the sum. I've got this sum of a sum is the sum of 56 00:04:05,350 --> 00:04:09,174 the sums. And I've got the derivative of a sum is 57 00:04:09,174 --> 00:04:13,836 the sum of derivatives. Fundamentally, mathematics is not just 58 00:04:13,836 --> 00:04:18,841 about these rules, right? It's about the relationships between all 59 00:04:18,841 --> 00:04:22,612 of these rules. We're seeing various objects now, 60 00:04:22,612 --> 00:04:26,092 integrals, derivatives, the sigma notation. 61 00:04:26,092 --> 00:04:30,303 And they're all sharing some common rules, right? 62 00:04:30,303 --> 00:04:41,833 And working out those relationships is really part of the fun.