1 00:00:00,000 --> 00:00:00,990 2 00:00:00,990 --> 00:00:05,310 In the last video we took essentially the length of y 3 00:00:05,309 --> 00:00:09,619 equals x square, not the length, but we went from zero 4 00:00:09,619 --> 00:00:14,339 to 1 on the x-axis and you can view it as this area. 5 00:00:14,339 --> 00:00:17,710 And we rotated around the y-axis to get this figure here 6 00:00:17,710 --> 00:00:19,470 and we figured out the volume, I think our answer if I 7 00:00:19,469 --> 00:00:21,329 remember was pi over 2. 8 00:00:21,329 --> 00:00:24,159 And we use what I called and what everyone calls 9 00:00:24,160 --> 00:00:25,469 the Shell method. 10 00:00:25,469 --> 00:00:27,089 I want to show you that you could actually used 11 00:00:27,089 --> 00:00:28,739 the disk method here. 12 00:00:28,739 --> 00:00:30,879 But then we'll just have to switch the ys and xs. 13 00:00:30,879 --> 00:00:36,229 Instead of writing this the function as a function of x, 14 00:00:36,229 --> 00:00:38,500 we'll just take the inverse of it and write it 15 00:00:38,500 --> 00:00:39,320 as a function of y. 16 00:00:39,320 --> 00:00:42,270 so this curve y equals x squared, what it can also be 17 00:00:42,270 --> 00:00:43,900 written as, we can just take the squared root of both 18 00:00:43,899 --> 00:00:48,619 sides as x is equal to the square root of y. 19 00:00:48,619 --> 00:00:50,629 Just take the square root of both sides. 20 00:00:50,630 --> 00:00:53,440 And now we can use this information to do the disk 21 00:00:53,439 --> 00:00:56,649 method, but everywhere where we had an x in the past 22 00:00:56,649 --> 00:00:57,500 we'll now have a y. 23 00:00:57,500 --> 00:00:59,560 So let's think about how we would do it. 24 00:00:59,560 --> 00:01:04,240 Once again it's this area-- really the hardest thing about 25 00:01:04,239 --> 00:01:07,109 all of these problems is just the visualization. 26 00:01:07,109 --> 00:01:08,040 I think that's why they do it. 27 00:01:08,040 --> 00:01:10,230 Just to make sure that you know how to visualize things and 28 00:01:10,230 --> 00:01:11,560 maybe understand the calculus. 29 00:01:11,560 --> 00:01:14,490 It's more visualization then calculus really. 30 00:01:14,489 --> 00:01:18,059 So we're still dealing with-- if we take a cross section, if 31 00:01:18,060 --> 00:01:23,269 we were to cut this figure this area would be the 32 00:01:23,269 --> 00:01:23,884 cross section. 33 00:01:23,885 --> 00:01:26,570 34 00:01:26,569 --> 00:01:28,929 And we're still rotating around the y-axis just like we 35 00:01:28,930 --> 00:01:29,900 did in the last video. 36 00:01:29,900 --> 00:01:32,290 We're still rotation around the y-axis, we get this figure. 37 00:01:32,290 --> 00:01:34,420 So how would we deal with disks. 38 00:01:34,420 --> 00:01:39,266 Well a disk would look like this at some point-- let 39 00:01:39,266 --> 00:01:41,690 me pick another color. 40 00:01:41,689 --> 00:01:46,539 At some point we'd have a disk like that. 41 00:01:46,540 --> 00:01:50,330 That would be the top of the disk, and it would have some 42 00:01:50,329 --> 00:01:53,519 depth like we did before. 43 00:01:53,519 --> 00:01:58,069 44 00:01:58,069 --> 00:02:02,309 And its height, or the radius of that disk, 45 00:02:02,310 --> 00:02:06,219 would be equal to x. 46 00:02:06,219 --> 00:02:07,230 It's equal to x. 47 00:02:07,230 --> 00:02:08,640 I know you're thinking it looks like the shell method. 48 00:02:08,639 --> 00:02:09,979 But what is x equal to? 49 00:02:09,979 --> 00:02:12,129 It's equal to the square root of y. 50 00:02:12,129 --> 00:02:14,159 It equals the square root of y. 51 00:02:14,159 --> 00:02:16,810 And what would be the width of that disk? 52 00:02:16,810 --> 00:02:19,400 Now we are making everything as a function of y, so the width 53 00:02:19,400 --> 00:02:22,430 would be just a very small distance d y, the 54 00:02:22,430 --> 00:02:23,420 differential y. 55 00:02:23,419 --> 00:02:28,030 56 00:02:28,030 --> 00:02:29,500 That's essentially all we need to know. 57 00:02:29,500 --> 00:02:32,870 So the volume of that disk would be the radius squared 58 00:02:32,870 --> 00:02:36,670 times pi times d y. 59 00:02:36,669 --> 00:02:40,069 Hopefully that makes a little bit sense, but there's another 60 00:02:40,069 --> 00:02:41,069 hitch on this problem. 61 00:02:41,069 --> 00:02:43,199 This is actually similar to what we did two videos ago. 62 00:02:43,199 --> 00:02:48,750 Because when you view it in the y-axis, from this y frame of 63 00:02:48,750 --> 00:02:51,439 reference, what we're going to do is we're going to take the 64 00:02:51,439 --> 00:02:57,659 volume, we take the volume of x is equal to 1. 65 00:02:57,659 --> 00:03:00,189 So what would be the volume of x is equal to 1 rotated 66 00:03:00,189 --> 00:03:01,099 around the x-axis? 67 00:03:01,099 --> 00:03:03,932 It would just be the entire cylinder. 68 00:03:03,932 --> 00:03:06,930 69 00:03:06,930 --> 00:03:08,409 It's really important that you visualize this 70 00:03:08,409 --> 00:03:10,780 right, what we're doing. 71 00:03:10,780 --> 00:03:15,400 We're going to figure out this volume, volume 72 00:03:15,400 --> 00:03:16,740 of x is equal to 1. 73 00:03:16,740 --> 00:03:21,030 74 00:03:21,030 --> 00:03:23,990 Let me draw the axis just so you know what we're doing. 75 00:03:23,990 --> 00:03:30,240 So this would be the y-axis, that would be the x-axis 76 00:03:30,240 --> 00:03:31,219 as best as I can draw. 77 00:03:31,219 --> 00:03:34,009 78 00:03:34,009 --> 00:03:35,659 Originally, we can figure out the volume of 79 00:03:35,659 --> 00:03:37,240 the entire cylinder. 80 00:03:37,240 --> 00:03:43,740 This is x is equal to 1, x is equal to 1 from what y points? 81 00:03:43,740 --> 00:03:44,960 From the point, well what is this? 82 00:03:44,960 --> 00:03:47,760 This is y equals 1. 83 00:03:47,759 --> 00:03:49,929 y equal 1 to zero. 84 00:03:49,930 --> 00:03:54,230 So we would figure out this volume from y equals 1 to zero. 85 00:03:54,229 --> 00:03:56,000 And how would we do that? 86 00:03:56,000 --> 00:03:58,750 What would be the integral? 87 00:03:58,750 --> 00:04:01,389 Remember everything is in y now, so it might seem 88 00:04:01,389 --> 00:04:03,659 a little confusing. 89 00:04:03,659 --> 00:04:07,180 For each of these disks, this is going to be made up of 90 00:04:07,180 --> 00:04:09,860 a bunch of disks, let me draw one of them. 91 00:04:09,860 --> 00:04:12,060 Let me draw the top one. 92 00:04:12,060 --> 00:04:16,170 The top disk, and it has a width, it's not going to be 93 00:04:16,170 --> 00:04:18,600 this entire cylinder, its width is d y, its width 94 00:04:18,600 --> 00:04:21,410 is just going to be this. 95 00:04:21,410 --> 00:04:24,560 d y, a very small sliver. 96 00:04:24,560 --> 00:04:32,160 d y, and its radius is x or you could say f of y. 97 00:04:32,160 --> 00:04:35,660 98 00:04:35,660 --> 00:04:38,040 What would be the volume of that disk? 99 00:04:38,040 --> 00:04:40,290 What would be the surface area of the top? 100 00:04:40,290 --> 00:04:44,350 It would be f of y squared. 101 00:04:44,350 --> 00:04:46,740 Remember we're dealing everything with y. f of y 102 00:04:46,740 --> 00:04:50,410 squared times pi, that gives the area of the top. 103 00:04:50,410 --> 00:04:54,510 Put the pi outside of the integral times d y. 104 00:04:54,509 --> 00:04:58,699 d y, and we said y is going from zero to 1. 105 00:04:58,699 --> 00:05:00,800 y is going from zero to 1. 106 00:05:00,800 --> 00:05:03,129 So that's that entire cylinder. 107 00:05:03,129 --> 00:05:05,860 And what we'll want to do is subtract out, essentially cut 108 00:05:05,860 --> 00:05:09,009 out the volume of the inner bowl. 109 00:05:09,009 --> 00:05:10,730 So minus. 110 00:05:10,730 --> 00:05:12,580 How would we figure out the inner bowl? 111 00:05:12,579 --> 00:05:16,409 That's where we will deal with the x is equal to the 112 00:05:16,410 --> 00:05:19,240 square root of y function. 113 00:05:19,240 --> 00:05:20,650 Because here, what's each disk? 114 00:05:20,649 --> 00:05:22,829 Once again it's f of y. 115 00:05:22,829 --> 00:05:24,979 I wrote this generally, this is f of y of the outside. 116 00:05:24,980 --> 00:05:27,580 117 00:05:27,579 --> 00:05:29,889 I'm going to now do it for the inside. 118 00:05:29,889 --> 00:05:32,979 I'm going to cut out the inside volume and because I kept 119 00:05:32,980 --> 00:05:34,160 it general I could do it. 120 00:05:34,160 --> 00:05:36,360 We're still going from zero to 1, from y equals 121 00:05:36,360 --> 00:05:37,639 zero to y equals 1. 122 00:05:37,639 --> 00:05:40,069 Remember we're doing with y now. 123 00:05:40,069 --> 00:05:47,490 I will take f of the inside f of y squared d of y. 124 00:05:47,490 --> 00:05:50,720 You can imagine in this case the inside function, we're 125 00:05:50,720 --> 00:05:51,850 going to take the volume. 126 00:05:51,850 --> 00:05:53,890 That original disk I drew is actually one of the disks 127 00:05:53,889 --> 00:05:55,259 for the inside function. 128 00:05:55,259 --> 00:06:00,639 Where the height of that disk is d y, that should be d y. 129 00:06:00,639 --> 00:06:03,050 The radius of the disk is f of y. 130 00:06:03,050 --> 00:06:06,259 And of course the area of the top of the desk is pi 131 00:06:06,259 --> 00:06:08,159 times the radius squared. 132 00:06:08,160 --> 00:06:10,840 How do we apply it to this particular situation? 133 00:06:10,839 --> 00:06:12,139 What is the outside function? 134 00:06:12,139 --> 00:06:15,789 f of y, x is equal to f of y. 135 00:06:15,790 --> 00:06:17,370 We are just switching the variables. 136 00:06:17,370 --> 00:06:18,840 We say that x is equal to 1. 137 00:06:18,839 --> 00:06:20,589 This is just a big cylinder, right? 138 00:06:20,589 --> 00:06:23,305 So f of i is equal go to 1 in this function, so we get pi-- 139 00:06:23,305 --> 00:06:28,419 let me switch colors-- times from zero to 1. 140 00:06:28,420 --> 00:06:35,530 1 squared d y minus pi, and we're still going 141 00:06:35,529 --> 00:06:38,169 from zero to 1. 142 00:06:38,170 --> 00:06:41,050 Remember, that's the y boundaries. 143 00:06:41,050 --> 00:06:42,569 What's f of y? 144 00:06:42,569 --> 00:06:51,540 Here f of y is square root of y squared d y. 145 00:06:51,540 --> 00:06:55,330 146 00:06:55,329 --> 00:06:56,439 Let's take the pi out. 147 00:06:56,439 --> 00:07:00,009 148 00:07:00,009 --> 00:07:04,279 So pi times, well 1 squared is 1. 149 00:07:04,279 --> 00:07:06,059 What's the anti-derivative of 1? 150 00:07:06,060 --> 00:07:08,579 What function's derivative is 1? 151 00:07:08,579 --> 00:07:11,370 It's x, right? 152 00:07:11,370 --> 00:07:13,109 x, that's the anti-derivative there. 153 00:07:13,110 --> 00:07:14,600 We can merge these. 154 00:07:14,600 --> 00:07:16,120 We took the pi out. 155 00:07:16,120 --> 00:07:20,209 This square root of y squared, this is just y. 156 00:07:20,209 --> 00:07:21,879 Sorry, we have that derivative 1. 157 00:07:21,879 --> 00:07:23,779 See, even I get quite confused here. 158 00:07:23,779 --> 00:07:26,549 We're taking this with respect to y, right? 159 00:07:26,550 --> 00:07:29,199 The anti-derivative of 1 now is y. 160 00:07:29,199 --> 00:07:31,930 We are setting up a bunch of ys. 161 00:07:31,930 --> 00:07:32,959 I'm sorry. 162 00:07:32,959 --> 00:07:34,699 I find this even a little perplexing when I start 163 00:07:34,699 --> 00:07:35,579 switching x and y. 164 00:07:35,579 --> 00:07:38,349 165 00:07:38,350 --> 00:07:39,980 The second function. 166 00:07:39,980 --> 00:07:41,540 Remember, we can merge these because it's the same 167 00:07:41,540 --> 00:07:43,520 boundaries and they're both in respect to y. 168 00:07:43,519 --> 00:07:45,329 Square root of y squared is y. 169 00:07:45,329 --> 00:07:49,939 What's the anti-derivative of y, the function y or y d y. 170 00:07:49,939 --> 00:07:57,319 Was y squared over 2, minus y squared over 2. 171 00:07:57,319 --> 00:08:00,490 We are going to evaluate that at 1 and zero. 172 00:08:00,490 --> 00:08:03,160 173 00:08:03,160 --> 00:08:04,000 So what does that get us? 174 00:08:04,000 --> 00:08:12,680 We get pi-- we get 1 minus 1/2 minus zero minus zero 175 00:08:12,680 --> 00:08:15,290 minus zero plus 0/2. 176 00:08:15,290 --> 00:08:16,600 Whatever these are zero. 177 00:08:16,600 --> 00:08:21,870 1 minus 1/2 that's 1/2 and times pi is 178 00:08:21,870 --> 00:08:23,540 equal to pie over 2. 179 00:08:23,540 --> 00:08:24,930 Which is the exact same result. 180 00:08:24,930 --> 00:08:28,009 And I was worried, because you never know when you might make 181 00:08:28,009 --> 00:08:29,939 a careless mistake as I often do. 182 00:08:29,939 --> 00:08:32,649 We got the exact same result as I got in the previous video 183 00:08:32,649 --> 00:08:34,850 when we used the shell method. 184 00:08:34,850 --> 00:08:37,050 The hardest thing here is just remembering that you're doing 185 00:08:37,049 --> 00:08:40,079 everything in terms of y. 186 00:08:40,080 --> 00:08:41,710 When we did the disk method traditionally, the disks 187 00:08:41,710 --> 00:08:43,050 were vertical disks. 188 00:08:43,049 --> 00:08:45,929 Now they're horizontal disks but it's the exact same thing. 189 00:08:45,929 --> 00:08:48,899 You just have to get your brain around the idea that we're 190 00:08:48,899 --> 00:08:51,779 dealing with the ys, that the boundaries on the integrals 191 00:08:51,779 --> 00:08:53,360 are now y values. 192 00:08:53,360 --> 00:08:58,629 We're taking the width of the disks, or the height of the 193 00:08:58,629 --> 00:09:01,889 disks are d y and the radius is now the function of y. 194 00:09:01,889 --> 00:09:05,730 Hopefully I didn't confuse you too much. 195 00:09:05,730 --> 00:09:07,889 My own brain malfunctioned a bit when I took the 196 00:09:07,889 --> 00:09:11,740 anti-derivative of 1 d y, it should have been y. 197 00:09:11,740 --> 00:09:13,899 I will see you in the next video where we will do even 198 00:09:13,899 --> 00:09:16,289 more complicated problems, where I'm sure I'll make 199 00:09:16,289 --> 00:09:16,969 even more mistakes. 200 00:09:16,970 --> 00:09:18,710 See you soon.