1 00:00:00,000 --> 00:00:00,820 2 00:00:00,820 --> 00:00:03,599 Where I left off, we were just essentially chugging through 3 00:00:03,600 --> 00:00:08,109 this fairly hairy derivative-- this definite integral-- 4 00:00:08,109 --> 00:00:08,910 this antiderivative. 5 00:00:08,910 --> 00:00:10,500 It takes my brain a little while to come 6 00:00:10,500 --> 00:00:11,150 up with the next term. 7 00:00:11,150 --> 00:00:14,300 So we evaluated at 2 and now we have to subtract 8 00:00:14,300 --> 00:00:16,320 this evaluated at 1. 9 00:00:16,320 --> 00:00:24,120 So minus 16 pi minus 4 pi over 3-- oh, sorry. 10 00:00:24,120 --> 00:00:24,650 Plus. 11 00:00:24,649 --> 00:00:26,389 The minus is going to be on everything. 12 00:00:26,390 --> 00:00:26,910 Oh no, sorry. 13 00:00:26,910 --> 00:00:28,269 That is a minus. 14 00:00:28,269 --> 00:00:29,289 That's a minus. 15 00:00:29,289 --> 00:00:31,869 Plus 4 pi. 16 00:00:31,870 --> 00:00:32,100 Right? 17 00:00:32,100 --> 00:00:33,380 Because x is 1. 18 00:00:33,380 --> 00:00:36,219 Minus pi over 2. 19 00:00:36,219 --> 00:00:38,439 And now we can simplify it. 20 00:00:38,439 --> 00:00:40,140 Let's see what we can do. 21 00:00:40,140 --> 00:00:41,340 This is really a hairy problem. 22 00:00:41,340 --> 00:00:44,750 23 00:00:44,750 --> 00:00:51,679 The 16 pi minus 8 pi, that equals 8 pi. 24 00:00:51,679 --> 00:00:53,030 And then that's a plus. 25 00:00:53,030 --> 00:00:55,859 The 32 plus 8 pi. 26 00:00:55,859 --> 00:00:59,479 32 pi plus 8 pi, that equals 40 pi. 27 00:00:59,479 --> 00:01:06,109 So let's see, I've simplified it to 40 pi, and what's minus 28 00:01:06,109 --> 00:01:08,739 8 times 4 is 32 pi over 3. 29 00:01:08,739 --> 00:01:13,829 30 00:01:13,829 --> 00:01:16,030 And then all of that, let's see if I can simplify this. 31 00:01:16,030 --> 00:01:18,989 Let's see, 16 pi plus 4 pi, that's 20 pi. 32 00:01:18,989 --> 00:01:22,559 33 00:01:22,560 --> 00:01:23,930 And then, let's see. 34 00:01:23,930 --> 00:01:28,060 Minus 4 pi over 3 minus pi over 2. 35 00:01:28,060 --> 00:01:30,740 So let's get a common denominator for this 36 00:01:30,739 --> 00:01:31,479 part right here. 37 00:01:31,480 --> 00:01:36,140 So if I put everything over 6-- 20 pi over 6 is the same thing 38 00:01:36,140 --> 00:01:45,015 is 120 pi over 6, and then minus-- 4 pi over 3, if I put 39 00:01:45,015 --> 00:01:48,890 it over 6 it becomes 8 pi over 6, right? 40 00:01:48,890 --> 00:01:53,060 And then pi over 2, if I put it over 6 it becomes 3 pi over 6. 41 00:01:53,060 --> 00:01:56,000 So minus 3 pi. 42 00:01:56,000 --> 00:01:58,239 So this whole term that we're going to have to subtract from 43 00:01:58,239 --> 00:02:04,329 this term is equal to 120 minus 11, right? 44 00:02:04,329 --> 00:02:07,260 So that's 109 pi over 6. 45 00:02:07,260 --> 00:02:09,080 This equals 1-- is that right? 46 00:02:09,080 --> 00:02:11,310 Yeah, 109 pi over 6. 47 00:02:11,310 --> 00:02:12,979 See what happens when you make up problems on the fly? 48 00:02:12,979 --> 00:02:14,500 You get ugly numbers. 49 00:02:14,500 --> 00:02:15,889 109 pi over 6. 50 00:02:15,889 --> 00:02:17,969 And what does that top part translate? 51 00:02:17,969 --> 00:02:19,000 So this is what we're going to subtract. 52 00:02:19,000 --> 00:02:22,955 This is when we evaluated the antiderivative at 1. 53 00:02:22,955 --> 00:02:24,510 And let's simplify this one. 54 00:02:24,509 --> 00:02:29,310 So that one, if we put 3 common denominator, that's 120 pi 55 00:02:29,310 --> 00:02:33,189 over 3 minus 32 pi over 3. 56 00:02:33,189 --> 00:02:38,609 120 minus 32, let's see, we get 90-- 88? 57 00:02:38,610 --> 00:02:39,290 Right. 58 00:02:39,289 --> 00:02:40,679 88 pi over 3. 59 00:02:40,680 --> 00:02:44,990 So that equals 88 pi over 3, and remember, that's 60 00:02:44,990 --> 00:02:46,320 just the top part. 61 00:02:46,319 --> 00:02:48,639 And then-- this is more of a review of fractions than 62 00:02:48,639 --> 00:02:50,819 anything else-- and then if I want to put it over 63 00:02:50,819 --> 00:02:52,829 6, I just double it. 64 00:02:52,830 --> 00:02:54,590 So I think we're almost done. 65 00:02:54,590 --> 00:02:57,229 Let me switch colors. 66 00:02:57,229 --> 00:02:59,789 So if you go over a denominator of 6-- 88 67 00:02:59,789 --> 00:03:01,099 pi over 3-- let's see. 68 00:03:01,099 --> 00:03:06,669 If I double that, I get 160-- 176, right? 69 00:03:06,669 --> 00:03:11,780 176 pi minus this. 70 00:03:11,780 --> 00:03:13,909 Minus 109 pi. 71 00:03:13,909 --> 00:03:16,210 I'm sure I made a careless mistake. 72 00:03:16,210 --> 00:03:19,099 These are my least favorite type of things to do. 73 00:03:19,099 --> 00:03:21,049 Hairy fractions. 74 00:03:21,050 --> 00:03:23,430 So 176 minus 109. 75 00:03:23,430 --> 00:03:26,000 That's the same thing as 76 minus 9. 76 00:03:26,000 --> 00:03:28,729 And that is 65. 77 00:03:28,729 --> 00:03:32,889 So our final answer is 65 pi over 6. 78 00:03:32,889 --> 00:03:36,069 Which isn't as hairy as it seemed when we 79 00:03:36,069 --> 00:03:36,689 started the problem. 80 00:03:36,689 --> 00:03:37,409 But that's pretty neat. 81 00:03:37,409 --> 00:03:39,789 This is kind of a strange shape. 82 00:03:39,789 --> 00:03:45,500 It's kind of a wide ring that has-- the upper part of the 83 00:03:45,500 --> 00:03:48,159 ring and the inner part of the ring are hard edges. 84 00:03:48,159 --> 00:03:50,090 But then it's curved on the outside. 85 00:03:50,090 --> 00:03:52,830 And we were able to figure out the volume of that. 86 00:03:52,830 --> 00:03:54,900 Especially-- and what was weird about this, is when it was 87 00:03:54,900 --> 00:04:00,349 rotated around the line y is equal to minus 2. 88 00:04:00,349 --> 00:04:01,739 Hopefully I didn't confuse you. 89 00:04:01,740 --> 00:04:06,120 These are about as difficult as these volume of 90 00:04:06,120 --> 00:04:09,060 revolution problems get. 91 00:04:09,060 --> 00:04:11,530 If you want more, just let me know. 92 00:04:11,530 --> 00:04:14,199 I will see you in the next video. 93 00:04:14,199 --> 00:04:14,399