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These are the user uploaded subtitles that are being translated: 1 00:00:00,350 --> 00:00:06,390 In this video we're talking about area under the curve versus area enclosed by the curve and the x axis 2 00:00:06,480 --> 00:00:09,390 and this is a less common application of definite integrals. 3 00:00:09,390 --> 00:00:13,920 Not all textbooks are going to ask you to make this distinction but if you run across a problem like 4 00:00:13,920 --> 00:00:16,790 this one this is probably what they're talking about. 5 00:00:16,920 --> 00:00:22,840 So area under the curve is by far the most common way to look at area. 6 00:00:22,860 --> 00:00:28,100 And when we talk about area under the curve it's also considered net area. 7 00:00:28,110 --> 00:00:30,090 So when you hear area under the curve. 8 00:00:30,090 --> 00:00:40,250 Think net area versus area enclosed by the curve and the x axis so enclosed would be a gross area. 9 00:00:40,250 --> 00:00:46,130 The same thing is a gross area and the difference is just with area under the curve or net area. 10 00:00:46,320 --> 00:00:52,390 You treat area above the x axis as positive an area below the x axis as negative. 11 00:00:52,560 --> 00:00:55,490 That's what you do when you take an integral of a function. 12 00:00:55,500 --> 00:01:01,470 Any area that's enclosed by the curve in the x axis that lies above the x axis is positive an area below 13 00:01:01,470 --> 00:01:03,350 the x axis is negative. 14 00:01:03,360 --> 00:01:05,920 So again because that's what we do when we integrate. 15 00:01:05,970 --> 00:01:08,580 That's by far the most common way to look at area. 16 00:01:08,760 --> 00:01:13,680 But sometimes you're going to hear this area enclosed by the curve and that means gross area and that's 17 00:01:13,680 --> 00:01:17,310 where retreat area above and below the x axis is positive. 18 00:01:17,310 --> 00:01:21,930 In other words we're taking absolute value of all of the area so we could right here. 19 00:01:21,930 --> 00:01:31,230 Area under the curve or net area we're going to be doing positive and negative area for area enclosed 20 00:01:31,260 --> 00:01:32,990 by the curb or gross area. 21 00:01:33,030 --> 00:01:36,960 We're going to be doing all positive area. 22 00:01:36,990 --> 00:01:38,160 So here's what that looks like. 23 00:01:38,160 --> 00:01:39,720 This is the information we've been given. 24 00:01:39,720 --> 00:01:42,210 We have this curve graphed in orange here. 25 00:01:42,210 --> 00:01:47,580 It's the curve f of x and notice at the interval for which it's defined as X equals negative 2 to X 26 00:01:47,580 --> 00:01:48,710 equals 6. 27 00:01:48,720 --> 00:01:52,890 We've also been given these three pieces of information here so we've been told that the integral of 28 00:01:52,890 --> 00:01:58,770 the function f of x so this function here in orange from negative to the positive one over that interval. 29 00:01:58,770 --> 00:02:03,960 So that's this interval right here negative to positive 1. 30 00:02:03,990 --> 00:02:07,220 So over this interval area is negative 2.8. 31 00:02:07,220 --> 00:02:12,870 And because this is the integral we're talking about area under the curve or net area where retreat 32 00:02:12,900 --> 00:02:15,390 area below the x axis as negative. 33 00:02:15,390 --> 00:02:22,050 So when this says negative 2.8 there acknowledging that this area from negative to positive one is all 34 00:02:22,050 --> 00:02:27,150 below the x axis and that the net area there is negative 2.8. 35 00:02:27,150 --> 00:02:31,740 Same thing here when we look at the integral of the function from 1 to 3 that means area under the curve 36 00:02:31,740 --> 00:02:32,790 or net area. 37 00:02:33,000 --> 00:02:35,750 So one to three is this interval. 38 00:02:35,760 --> 00:02:40,000 Right here from one to here three. 39 00:02:40,080 --> 00:02:42,020 So the area there is 1.2. 40 00:02:42,090 --> 00:02:47,310 It's positive because we can see that all of that area is above the x axis so it makes sense that they 41 00:02:47,310 --> 00:02:49,160 would indicate a positive area. 42 00:02:49,200 --> 00:02:54,760 Then this last integral here the integral from 1 to 6 is going to give us an area of negative 3.5. 43 00:02:54,810 --> 00:02:58,800 So before we're too quick to label this third area as negative 3.5. 44 00:02:58,920 --> 00:03:01,980 Notice that the interval here is 1 to 6. 45 00:03:01,980 --> 00:03:11,430 So that's the interval here from 1 all the way to 6 so this entire interval here we already know that 46 00:03:11,430 --> 00:03:15,420 that interval includes this positive area of 1.2. 47 00:03:15,630 --> 00:03:22,170 So if we go ahead and call this area right here a we know that taking the integral from to 5:59 looks 48 00:03:22,170 --> 00:03:27,570 at the net area or the area under the curve on this entire interval from one to six which means that 49 00:03:27,570 --> 00:03:30,910 it's going to treat this area between 1 and 3 as positive. 50 00:03:31,110 --> 00:03:34,430 And this area a between 3 and 6 as negative. 51 00:03:34,590 --> 00:03:42,210 So what we would want to say then is that positive 1.2 Plus a is going to be equal to the net total 52 00:03:42,450 --> 00:03:48,120 of negative three point five we would solve for a by subtracting 1.2 from both sides and we would get 53 00:03:48,180 --> 00:03:51,500 a is equal to negative four point seven. 54 00:03:51,510 --> 00:03:57,750 So we can say then that this area between 3 and six this entire area under the curve right here is negative 55 00:03:57,810 --> 00:03:59,040 four point seven. 56 00:03:59,130 --> 00:04:04,170 So in that area or area under the curve or the area you get when you take the integral when you take 57 00:04:04,170 --> 00:04:09,900 the integral if you get a negative result that means there's more area under the x axis than there is 58 00:04:09,930 --> 00:04:11,550 above the x axis. 59 00:04:11,550 --> 00:04:17,400 Similarly if you get a positive result that means there's more area above the x axis than below it. 60 00:04:17,430 --> 00:04:23,220 So now that we have our three areas if we want to find area under the curve over the entire interval 61 00:04:23,280 --> 00:04:28,030 negative to positive 6 the entire interval for the function we want to find. 62 00:04:28,030 --> 00:04:29,010 NET area. 63 00:04:29,010 --> 00:04:32,030 That means we take into account positive and negative values. 64 00:04:32,070 --> 00:04:40,410 So we would say that this is equal to negative 2.8 plus a positive 1.2 Plus a negative 4.7 or just minus 65 00:04:40,520 --> 00:04:41,620 4.7. 66 00:04:41,640 --> 00:04:45,490 And when we simplify here we get a negative 6.3. 67 00:04:45,540 --> 00:04:49,900 So that means the net area or the area under the curve is negative 6.3. 68 00:04:49,980 --> 00:04:56,160 That means that if we took the integral from negative to positive 6 we would get negative 6.3 because 69 00:04:56,160 --> 00:05:00,940 this net area is what we get when we take the integral and course it should make sense that we get a 70 00:05:00,940 --> 00:05:05,350 negative answer because if we look here at the graph we can see that we have more negative area more 71 00:05:05,350 --> 00:05:09,060 area under the x axis than we do above the x axis. 72 00:05:09,130 --> 00:05:15,130 If on the other hand we want to find area enclosed by the curve and the x axis or gross area we treat 73 00:05:15,160 --> 00:05:16,150 everything as positive. 74 00:05:16,150 --> 00:05:21,590 So we take the absolute value of each section of area that's enclosed by the curve and the x axis. 75 00:05:21,640 --> 00:05:28,860 So instead of negative 2.8 we say the absolute value of negative 2.8 Plus the absolute value of 1.2 76 00:05:28,960 --> 00:05:32,610 Plus the absolute value of negative four point seven. 77 00:05:32,620 --> 00:05:34,410 In other words we just treat everything as positive. 78 00:05:34,410 --> 00:05:41,640 So we end up with here is positive 2.8 plus a positive 1.2 plus a positive 4.7. 79 00:05:41,650 --> 00:05:46,690 And when we simplify that we get a positive eight point seven and that's the difference between area 80 00:05:46,750 --> 00:05:49,360 under the curve an area enclosed by the curve. 8887