1 00:00:00,000 --> 00:00:00,500 2 00:00:00,500 --> 00:00:04,279 In the last couple of videos we've been slowly moving 3 00:00:04,280 --> 00:00:07,580 towards our goal of figuring out the surface area 4 00:00:07,580 --> 00:00:09,210 of this torus. 5 00:00:09,210 --> 00:00:11,880 And we did it by evaluating a surface integral, and in order 6 00:00:11,880 --> 00:00:14,300 to evaluate a surface integral we had to take the 7 00:00:14,300 --> 00:00:17,530 parameterization-- take its partial with respect to s and 8 00:00:17,530 --> 00:00:18,830 with respect to t. 9 00:00:18,829 --> 00:00:20,589 We did that in the first video. 10 00:00:20,589 --> 00:00:22,410 Then we had to take its cross product. 11 00:00:22,410 --> 00:00:24,079 We did that in the second video. 12 00:00:24,079 --> 00:00:29,049 Now, we're ready to take the magnitude of the cross product. 13 00:00:29,050 --> 00:00:31,650 And then we can evaluate it inside of a double integral and 14 00:00:31,649 --> 00:00:35,119 we will have solved or we would have computed an actual surface 15 00:00:35,119 --> 00:00:37,609 integral-- something you see very few times in your 16 00:00:37,609 --> 00:00:38,369 education career. 17 00:00:38,369 --> 00:00:40,570 So this is kind of exciting. 18 00:00:40,570 --> 00:00:43,420 So this was the cross product right here. 19 00:00:43,420 --> 00:00:46,200 Now, let's take the magnitude of this thing. 20 00:00:46,200 --> 00:00:49,710 And you might remember, the magnitude of any vector is kind 21 00:00:49,710 --> 00:00:50,980 of a Pythagorean theorem. 22 00:00:50,979 --> 00:00:53,379 And in this case it's going to be kind of the distance 23 00:00:53,380 --> 00:00:56,400 formula to the Pythagorean theorem n 3 dimensions. 24 00:00:56,399 --> 00:01:01,759 So the magnitude-- this is equal to, just as a reminder, 25 00:01:01,759 --> 00:01:03,179 is equal to this right here. 26 00:01:03,179 --> 00:01:08,319 It's equal to the partial of r with respect to s cross 27 00:01:08,319 --> 00:01:11,250 with the partial of r with respect to t. 28 00:01:11,250 --> 00:01:13,549 Let me copy it and paste it. 29 00:01:13,549 --> 00:01:15,980 That is equal to that right there. 30 00:01:15,980 --> 00:01:17,890 Put an equal sign. 31 00:01:17,890 --> 00:01:19,689 These two quantities are equal. 32 00:01:19,689 --> 00:01:22,259 Now we want to figure out the magnitude. 33 00:01:22,260 --> 00:01:29,859 So if we want to take the magnitude of this thing, that's 34 00:01:29,859 --> 00:01:32,450 going to be equal to-- well, this is just a scalar that's 35 00:01:32,450 --> 00:01:33,570 multiply everything. 36 00:01:33,569 --> 00:01:36,039 So let's just write the scalar out there. 37 00:01:36,040 --> 00:01:42,540 So b plus a cosine of s times the magnitude of 38 00:01:42,540 --> 00:01:44,310 this thing right here. 39 00:01:44,310 --> 00:01:49,950 And the magnitude of this thing right here is going to be the 40 00:01:49,950 --> 00:01:53,280 sum, of-- you can imagine, it's the square root of this 41 00:01:53,280 --> 00:01:55,510 thing dotted with itself. 42 00:01:55,510 --> 00:01:58,500 Or you could say it's the sum of the squares of each of 43 00:01:58,500 --> 00:02:01,030 these terms to the 1/2 power. 44 00:02:01,030 --> 00:02:03,799 So let me write it like that. 45 00:02:03,799 --> 00:02:06,329 Let me write the sum of the squares. 46 00:02:06,329 --> 00:02:14,159 So if you square this you get a squared cosine squared 47 00:02:14,159 --> 00:02:17,539 of s, sine squared of t. 48 00:02:17,539 --> 00:02:19,169 That's that term. 49 00:02:19,169 --> 00:02:20,699 Plus-- let me color code it. 50 00:02:20,699 --> 00:02:23,530 That's that term. 51 00:02:23,530 --> 00:02:24,930 I'll do the magenta. 52 00:02:24,930 --> 00:02:27,200 Plus that term squared. 53 00:02:27,199 --> 00:02:36,229 Plus a squared cosine squared of s, cosine squared of t. 54 00:02:36,229 --> 00:02:37,500 That's that term. 55 00:02:37,500 --> 00:02:40,580 And then finally-- I'll do another color-- 56 00:02:40,580 --> 00:02:41,450 this term squared. 57 00:02:41,449 --> 00:02:47,899 So plus a squared sine squared of s. 58 00:02:47,900 --> 00:02:52,480 And it's going to be all of this business to the 1/2 power. 59 00:02:52,479 --> 00:02:57,209 This right here is the same thing as the magnitude 60 00:02:57,210 --> 00:02:58,310 of this right here. 61 00:02:58,310 --> 00:03:01,170 This is just a scalar that's multiplying by 62 00:03:01,169 --> 00:03:02,139 both of these terms. 63 00:03:02,139 --> 00:03:04,589 So let's see if we can do anything interesting here. 64 00:03:04,590 --> 00:03:06,080 If this can be simplified in any way. 65 00:03:06,080 --> 00:03:08,590 We have a squared cosine squared of s. 66 00:03:08,590 --> 00:03:11,830 We have an a squared cosine squared of s here, so let's 67 00:03:11,830 --> 00:03:15,360 factor that out from both of these terms and 68 00:03:15,360 --> 00:03:17,280 see what happens. 69 00:03:17,280 --> 00:03:19,219 I'm just going to rewrite this second part. 70 00:03:19,219 --> 00:03:23,949 So this is going to be a squared cosine squared of s 71 00:03:23,949 --> 00:03:34,099 times sine squared of t-- put a parentheses-- plus cosine-- 72 00:03:34,099 --> 00:03:37,789 oh, I want to do it in that magenta color, not orange. 73 00:03:37,789 --> 00:03:40,969 Plus cosine squared of t. 74 00:03:40,969 --> 00:03:44,900 75 00:03:44,900 --> 00:03:50,330 And then you're going to have this plus a squared 76 00:03:50,330 --> 00:03:52,590 sine squared of s. 77 00:03:52,590 --> 00:03:59,810 And of course, all of that is to the 1/2 power. 78 00:03:59,810 --> 00:04:00,789 Now what is this? 79 00:04:00,789 --> 00:04:03,840 Well, we have sine squared of t plus cosine squared of t. 80 00:04:03,840 --> 00:04:04,180 That's nice. 81 00:04:04,180 --> 00:04:08,319 That's equal to 1, the most basic of trig identities. 82 00:04:08,319 --> 00:04:14,169 So this expression right here simplifies to a squared cosine 83 00:04:14,169 --> 00:04:21,339 squared of s plus this over here: a squared 84 00:04:21,339 --> 00:04:23,779 sine squared of s. 85 00:04:23,779 --> 00:04:26,449 And all of that to the 1/2 power. 86 00:04:26,449 --> 00:04:28,250 You might immediately recognize you can factor 87 00:04:28,250 --> 00:04:29,569 out an a squared. 88 00:04:29,569 --> 00:04:34,159 This is equal to a squared times cosine squared of s 89 00:04:34,160 --> 00:04:40,880 plus sine squared of s. 90 00:04:40,879 --> 00:04:43,659 And all of that to the 1/2 power. 91 00:04:43,660 --> 00:04:45,340 I'm just focusing on this term right here. 92 00:04:45,339 --> 00:04:46,810 I'll write this in a second. 93 00:04:46,810 --> 00:04:49,519 But once again, cosine squared plus sine squared of anything 94 00:04:49,519 --> 00:04:51,939 is going to be equal 1 as long as it's the same anything 95 00:04:51,939 --> 00:04:52,930 it's equal to 1. 96 00:04:52,930 --> 00:04:56,079 So this term is a squared to the 1/2 power. 97 00:04:56,079 --> 00:04:58,849 Or the square root of a squared, which is just 98 00:04:58,850 --> 00:05:00,379 going to be equal to a. 99 00:05:00,379 --> 00:05:04,219 So all of this-- all that crazy business right here just 100 00:05:04,220 --> 00:05:08,340 simplifies, all of that just simplifies to a. 101 00:05:08,339 --> 00:05:12,579 So this cross product here simplifies to this times 102 00:05:12,579 --> 00:05:16,259 a, which is a pretty neat and clean simplification. 103 00:05:16,259 --> 00:05:17,394 So let me rewrite this. 104 00:05:17,394 --> 00:05:21,089 105 00:05:21,089 --> 00:05:30,039 That simplifies, it simplifies to a times that. 106 00:05:30,040 --> 00:05:33,100 And what's that? a times b, so it's ab. 107 00:05:33,100 --> 00:05:36,520 ab plus a squared cosine of s. 108 00:05:36,519 --> 00:05:40,549 109 00:05:40,550 --> 00:05:43,020 So already, we've gotten pretty far and it's nice when you do 110 00:05:43,019 --> 00:05:45,229 something so beastly and eventually it gets to 111 00:05:45,230 --> 00:05:48,590 something reasonably simple. 112 00:05:48,589 --> 00:05:51,849 And just to review what we had to do, what our mission was 113 00:05:51,850 --> 00:05:55,710 several videos ago, is we want to evaluate what this thing is 114 00:05:55,709 --> 00:06:00,109 over the region from s-- over the region over with the 115 00:06:00,110 --> 00:06:01,090 surface is defined. 116 00:06:01,089 --> 00:06:06,459 So s going from 0 to 2 pi and t going from 0 to 2 pi. 117 00:06:06,459 --> 00:06:07,889 Over this region. 118 00:06:07,889 --> 00:06:10,860 So we want to integrate this over that region. 119 00:06:10,860 --> 00:06:16,720 So that region we're going to vary s from 0 to 2 pi. 120 00:06:16,720 --> 00:06:18,130 So ds. 121 00:06:18,129 --> 00:06:23,870 And then we're going to vary t from 0 to 2 pi-- dt. 122 00:06:23,870 --> 00:06:25,459 And this is what we're evaluating. 123 00:06:25,459 --> 00:06:28,370 We're evaluating the magnitude of the cross product of these 124 00:06:28,370 --> 00:06:32,060 two partial derivatives of our original parameterization. 125 00:06:32,060 --> 00:06:33,920 So this is what we can put in there. 126 00:06:33,920 --> 00:06:36,009 Things are getting simple all of a sudden, or simpler. 127 00:06:36,009 --> 00:06:41,180 ab plus a squared cosine of s. 128 00:06:41,180 --> 00:06:43,079 And what is this equal to? 129 00:06:43,079 --> 00:06:47,430 So this is going to be equal to-- well, we just take 130 00:06:47,430 --> 00:06:51,610 antiderivative of the inside with respect to s. 131 00:06:51,610 --> 00:06:53,800 So the antiderivative-- so let me do the outside 132 00:06:53,800 --> 00:06:54,829 of our integral. 133 00:06:54,829 --> 00:06:56,689 So we're still going to have to deal with the 0 to 2 134 00:06:56,689 --> 00:07:00,589 pi and our dt right here. 135 00:07:00,589 --> 00:07:03,899 But the antiderivative with respect to s right here is 136 00:07:03,899 --> 00:07:06,629 going to be-- ab is just a constant, so it's going to 137 00:07:06,629 --> 00:07:11,920 be abs plus-- what's the antiderivative of cosine of s? 138 00:07:11,920 --> 00:07:12,840 It's sine of s. 139 00:07:12,839 --> 00:07:17,539 So plus a squared sine of s. 140 00:07:17,540 --> 00:07:21,860 And we're going to evaluate it from 0 to 2 pi. 141 00:07:21,860 --> 00:07:24,270 And what is this going to be equal to? 142 00:07:24,269 --> 00:07:27,899 Let's put our boundaries out again or the t integral that 143 00:07:27,899 --> 00:07:33,799 we're going to have to do in a second-- 0 the 2 pi d t. 144 00:07:33,800 --> 00:07:37,480 When you put 2 pi here you're going to get ab times 145 00:07:37,480 --> 00:07:39,240 2 pi or 2 pi ab. 146 00:07:39,240 --> 00:07:45,840 So you're going to have 2 pi ab plus a squared sine of 2 pi. 147 00:07:45,839 --> 00:07:48,669 Sine of 2 pi is 0, so there's not going to be any term there. 148 00:07:48,670 --> 00:07:54,750 And then minus 0 times ab, which is 0. 149 00:07:54,750 --> 00:07:57,220 And then you're going to have minus a squared sine 150 00:07:57,220 --> 00:07:58,620 of 0, which is also 0. 151 00:07:58,620 --> 00:08:00,300 So all of the other terms are all 0's. 152 00:08:00,300 --> 00:08:03,139 So that's what we're left with it, it simplified nicely. 153 00:08:03,139 --> 00:08:05,139 So now we just have to take the antiderivative of 154 00:08:05,139 --> 00:08:08,709 this with respect to t. 155 00:08:08,709 --> 00:08:11,620 And this is a constant in t, so this is going to be equal to-- 156 00:08:11,620 --> 00:08:17,750 take the antiderivative with respect to t-- 2 pi abt and we 157 00:08:17,750 --> 00:08:21,949 need to evaluate that from 0 to 2 pi, which is equal to-- 158 00:08:21,949 --> 00:08:23,569 so we put 2 pi in there. 159 00:08:23,569 --> 00:08:27,339 You have a 2 pi for t, it'll be a 2 pi times 2 pi ab. 160 00:08:27,339 --> 00:08:33,250 Or we should say, 2 pi squared times ab minus 161 00:08:33,250 --> 00:08:34,399 0 times this thing. 162 00:08:34,399 --> 00:08:35,879 Well, that's just going to be 0, so we don't even 163 00:08:35,879 --> 00:08:37,230 have to write it down. 164 00:08:37,230 --> 00:08:37,860 So we're done. 165 00:08:37,860 --> 00:08:40,639 This is the surface area of the torus. 166 00:08:40,639 --> 00:08:41,409 This is exciting. 167 00:08:41,409 --> 00:08:43,100 It just kind of snuck up on us. 168 00:08:43,100 --> 00:08:49,680 This is equal to 4 pi squared ab, which is kind of a neat 169 00:08:49,679 --> 00:08:51,750 formula because it's very neat and clean. 170 00:08:51,750 --> 00:08:53,730 You know, it has a 2 pi, which is kind of the 171 00:08:53,730 --> 00:08:55,720 diameter of a circle. 172 00:08:55,720 --> 00:08:57,860 We're squaring it, which kind of makes sense because we're 173 00:08:57,860 --> 00:09:03,629 taking the product of-- you can kind of imagine the product 174 00:09:03,629 --> 00:09:04,700 of these 2 circles. 175 00:09:04,700 --> 00:09:07,150 I'm speaking in very abstract, general terms, but that 176 00:09:07,149 --> 00:09:08,379 kind of feels good. 177 00:09:08,379 --> 00:09:09,909 And then we're taking just the product of those 178 00:09:09,909 --> 00:09:11,339 two radiuses, remember. 179 00:09:11,340 --> 00:09:14,310 Let me just copy this thing down here. 180 00:09:14,309 --> 00:09:17,159 Actually, let me copy this thing because this is our new-- 181 00:09:17,159 --> 00:09:19,569 this is our exciting result. 182 00:09:19,570 --> 00:09:20,710 Let me copy this. 183 00:09:20,710 --> 00:09:23,110 So copy. 184 00:09:23,110 --> 00:09:27,350 So all of this work that we did simplified to this, 185 00:09:27,350 --> 00:09:30,460 which is exciting. 186 00:09:30,460 --> 00:09:33,990 We now know that if you have a torus where the radius of the 187 00:09:33,990 --> 00:09:40,440 cross section is a, and the radius from the center of 188 00:09:40,440 --> 00:09:44,130 the torus to the middle of the cross sections is b. 189 00:09:44,129 --> 00:09:47,820 That the surface area of that torus is going to be 4 pi 190 00:09:47,820 --> 00:09:50,379 squared times a times b. 191 00:09:50,379 --> 00:09:53,689 Which I think is a pretty neat outcome. 192 00:09:53,690 --> 00:09:53,933