1 00:00:00,25 --> 00:00:08,650 [MUSIC] We're going to sneak up on a certain fact. 2 00:00:08,650 --> 00:00:11,320 I want to show that pi is less than 22 sevenths. 3 00:00:11,320 --> 00:00:15,735 These are awfully close. Pi is about 3.1416. 4 00:00:15,735 --> 00:00:17,330 And 22 seventh is about 3.1429. Alright? 5 00:00:17,330 --> 00:00:19,490 These are awfully close. And we'll do it with an[UNKNOWN]. 6 00:00:19,490 --> 00:00:26,216 I'm just going to make something up, and it'll turn out in retrospect to be 7 00:00:26,216 --> 00:00:31,949 exactly what we need. So the trick's going to be, do the 8 00:00:31,949 --> 00:00:39,882 integral from 0 to 1 of x to the fourth. Times one minus x to the fourth all over 9 00:00:39,882 --> 00:00:45,260 one plus x squared dx. How do I integrate that? 10 00:00:45,260 --> 00:00:48,284 Well, before we actually do the calculation, let's notice something about 11 00:00:48,284 --> 00:00:53,190 the integrand. the denominator here is positive. 12 00:00:53,190 --> 00:00:57,411 And the numerator, well as long as x isn't 0 or 1, the numerator's positive as 13 00:00:57,411 --> 00:01:00,892 well. It's continuous and I'm integrating a 14 00:01:00,892 --> 00:01:05,640 continuous function that's mostly positive except at the end point. 15 00:01:05,640 --> 00:01:10,300 So that means that this integral ends up being positive as well. 16 00:01:10,300 --> 00:01:13,426 We'll need that fact later. The, the fact that the integral's 17 00:01:13,426 --> 00:01:16,62 positive. But let's work now on the integrand. 18 00:01:16,62 --> 00:01:19,0 I'm going to focus first on the numerator. 19 00:01:19,0 --> 00:01:24,908 1 minus x to the fourth. If I just multiply this out I get x to 20 00:01:24,908 --> 00:01:33,257 the fourth minus 4x cubed plus 6x squared minus 4x plus 1. 21 00:01:34,440 --> 00:01:39,965 And now if I multiply this by x to the fourth, that just adds 4 to each of these 22 00:01:39,965 --> 00:01:45,391 exponents. This will be x to the eighth minus 4 x to 23 00:01:45,391 --> 00:01:54,420 the seventh, plus 6 x to the sixth, minus 4 x to the fifth, plus x to the fourth. 24 00:01:54,420 --> 00:01:58,762 Let's get ready to do some long division. So I'm just going to copy this down. 25 00:01:58,762 --> 00:02:03,412 So, what did I get as the numerator, what was x to the fourth times 1 minus x to 26 00:02:03,412 --> 00:02:08,38 the fourth. Expand it out, was x to the eight minus 27 00:02:08,38 --> 00:02:13,910 4x to the seventh plus 6x to the sixth minus 4x to the fifth. 28 00:02:13,910 --> 00:02:17,273 Plus x to the fourth, and I'm going to take that numerator and I divide it by 29 00:02:17,273 --> 00:02:21,470 one plus x squared, so this we're going to do long division. 30 00:02:21,470 --> 00:02:26,938 I'm dividing this by, I write it as x squared plus one. 31 00:02:26,938 --> 00:02:29,0 Okay, so that's the calculation I want to do. 32 00:02:29,0 --> 00:02:32,271 I want to do this long division problem. Well I gotta put something up here that I 33 00:02:32,271 --> 00:02:35,420 multiply by x squared plus 1 to kill x to the 8th. 34 00:02:35,420 --> 00:02:39,450 And, for that I could use x to the 6th, because x to the 6th times x squared is x 35 00:02:39,450 --> 00:02:43,674 to the eighth. And x to the sixth times 1, which is plus 36 00:02:43,674 --> 00:02:47,216 x to the 6th. And I'm going to subtract that, and what 37 00:02:47,216 --> 00:02:50,178 do I get? Well, the x to the 8th terms cancel, and 38 00:02:50,178 --> 00:02:54,766 I've got a minus 4x to the 7th. 6x to the sixth minus x to the sixth is 5 39 00:02:54,766 --> 00:03:00,77 x to the sixth minus 4 x to the fifth and plus x to the fourth. 40 00:03:00,77 --> 00:03:04,550 Those survive and I'm subtracting anything as soon as copy them down, minus 41 00:03:04,550 --> 00:03:09,956 4 x to the fifth plus x to the fourth. Alright, now you gotta put something up 42 00:03:09,956 --> 00:03:15,430 here that I can multiply by x squared plus 1 to your minus 4 x to the seventh. 43 00:03:15,430 --> 00:03:20,464 but minus 4x to the fifth. because, that times x squared gives me 44 00:03:20,464 --> 00:03:27,50 minus 4x to the seventh and that times 1 gives me ooh, minus 4x to the fifth. 45 00:03:27,50 --> 00:03:29,890 That's Great! So this term dies and this term dies and 46 00:03:29,890 --> 00:03:34,450 let's copy down the terms actually it's 5x to the sixth. 47 00:03:34,450 --> 00:03:38,341 Plus x to the fourth. Now what do I put up here to multiply by 48 00:03:38,341 --> 00:03:42,42 this to kill this? plus five x to the fourth will do it, 49 00:03:42,42 --> 00:03:46,242 because that times x squared gives me five x to the sixth, and five x to the 50 00:03:46,242 --> 00:03:51,554 fourth times one gives me plus 5x to the fourth. 51 00:03:51,554 --> 00:03:56,0 And remember, I'm subtracting this. So that kills this first term. 52 00:03:56,0 --> 00:03:58,960 But here I've got what minus 4x to the fourth. 53 00:03:58,960 --> 00:04:02,590 I'm going to write that over here. So, minus 4x to the fourth is all that's 54 00:04:02,590 --> 00:04:04,856 left. I've gotta put something up here to 55 00:04:04,856 --> 00:04:08,977 multiply by this to kill this. How about minus 4x squared, because that 56 00:04:08,977 --> 00:04:13,270 times this gives me minus 4x to the fourth. 57 00:04:13,270 --> 00:04:19,100 And then minus 4x squared. I'm subtracting this so this becomes just 58 00:04:19,100 --> 00:04:22,210 4x squared. I've got to put something here to 59 00:04:22,210 --> 00:04:26,324 multiply by this to kill this. Well plus 4 will do it. 60 00:04:26,324 --> 00:04:31,240 Plus 4 times that gives me 4x squared plus 4. 61 00:04:31,240 --> 00:04:36,900 And when I subtract this I get minus four left over and that's my remainder. 62 00:04:36,900 --> 00:04:40,342 Now we can rewrite the integral. So in light of the expansion of the 63 00:04:40,342 --> 00:04:44,478 numerator and the long division after I divided by 1 plus x squared. 64 00:04:44,478 --> 00:04:50,50 This is the integral from 0 to 1. Of, well, what did I get when I divided? 65 00:04:50,50 --> 00:04:54,534 Was x to the sixth minus four x to the fifth plus five x to the fourth minus 66 00:04:54,534 --> 00:05:00,891 four x squared. Plus four and then I had a remainder so 67 00:05:00,891 --> 00:05:07,480 minus four over one plus x squared. This is an integral I can do. 68 00:05:07,480 --> 00:05:13,220 Okay this is x to the seventh over seven. That is the anti-derivative x to the 69 00:05:13,220 --> 00:05:17,345 sixth. the anti-derivative here is four x to the 70 00:05:17,345 --> 00:05:21,820 sixth over six. Plus an antiderivative here. 71 00:05:21,820 --> 00:05:29,487 It's just x to the fifth, antiderivative here is minus 4 x to the third over 3. 72 00:05:29,487 --> 00:05:33,455 Antiderivative of 4 is just plus 4x, and an antiderivative of 4 over 1 plus x 73 00:05:33,455 --> 00:05:39,400 squared, well the antiderivative of 1 over 1 plus x squared's arctan. 74 00:05:39,400 --> 00:05:45,841 So this is minus 4, orcten x and I'm integrating from 0 to 1, so I'm just 75 00:05:45,841 --> 00:05:55,110 going to evaluate this from 0 to 1. Now, what do I get when I evaluate this? 76 00:05:55,110 --> 00:06:01,481 When I plug in 0, I get 0, 0, 0, 0, 0. And arctan of 0 is 0. 77 00:06:01,481 --> 00:06:05,170 So, I end up just subtracting 0. So I only have to just evaluate this at 1 78 00:06:05,170 --> 00:06:08,220 to figure out what this integral is equal to. 79 00:06:08,220 --> 00:06:17,711 So when I plug in 1, I get 1 seventh. Minus 4 sixths plus one minus 4 thirds 80 00:06:17,711 --> 00:06:25,334 plus 4 minus 4 times, and what's arctan of 1, well arctan of is pi over 4, let's 81 00:06:25,334 --> 00:06:32,104 simplify a bit. So, I've got 1 seventh, and I'll put all 82 00:06:32,104 --> 00:06:38,240 these with a common denominator of three. So, minus 2 thirds plus 3 thirds minus 4 83 00:06:38,240 --> 00:06:46,243 thirds plus 12 thirds, and then minus 4 times pi over 4 is just minus pi. 84 00:06:47,600 --> 00:06:50,350 Well now what do I have? let's see here. 85 00:06:50,350 --> 00:06:55,372 So I've got minus 2 minus 4 that's minus 6 thirds, 3 plus 12, that's 15 thirds 86 00:06:55,372 --> 00:07:02,22 minus 6 thirds. That gives me 9 thirds leftover, and then 87 00:07:02,22 --> 00:07:05,796 also minus pi. So I've got a seventh. 88 00:07:05,796 --> 00:07:16,25 Plus, well 9 thirds is just 3 minus pi. Well 1 seventh plus 3 that's 22 sevenths 89 00:07:16,25 --> 00:07:20,664 minus pi. Here's the significance of this, remember 90 00:07:20,664 --> 00:07:24,288 back to the beginning. Well way back at the beginning I pointed 91 00:07:24,288 --> 00:07:28,620 out that this integral is positive. So what have we shown? 92 00:07:28,620 --> 00:07:32,790 So we've shown that this, the value of the integral is positive. 93 00:07:32,790 --> 00:07:38,476 In other words, that 22 sevenths is bigger than pi. 94 00:07:38,476 --> 00:07:41,442 So this is really a fun example. We've got all these pieces coming 95 00:07:41,442 --> 00:07:43,946 together. We've got the derivative of arctan, long 96 00:07:43,946 --> 00:07:47,900 division of polynomials, the fundamental theorem of calculus. 97 00:07:47,900 --> 00:07:53,716 All these forces combining to convince us that 22 sevenths is bigger than pi. 98 00:07:53,716 --> 00:07:57,636 So if you want an entertaining challenge, here is an even more complicated integral 99 00:07:57,636 --> 00:08:01,46 to evaluate. The interval x goes from 0 to 1 of x to 100 00:08:01,46 --> 00:08:07,731 the eighth of 1 minus x to the eighth. Times 25 plus 816 times x squared and all 101 00:08:07,731 --> 00:08:13,276 of that over 3164 times one plus x squared. 102 00:08:13,276 --> 00:08:25,133 And the answer that you get here is pretty entertaining.