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These are the user uploaded subtitles that are being translated: 1 00:00:02,030 --> 00:00:07,730 The integral function models the area underneath the graph of a function and it's calculated opposite 2 00:00:07,970 --> 00:00:14,900 of the way we calculated derivatives which is why an integral is also called an anti derivative instead 3 00:00:14,900 --> 00:00:18,100 of looking for the derivative of a function as we did before. 4 00:00:18,140 --> 00:00:24,200 We'll be looking for the function we would have differentiated to arrive at our current function. 5 00:00:24,220 --> 00:00:29,590 Think about it this way from what we know about derivatives we can see that the derivative of f of x 6 00:00:29,710 --> 00:00:34,640 is x and that the derivative of g of X is HMX. 7 00:00:34,750 --> 00:00:40,150 Previously we were always given g of X and only asked to calculate the derivative and we would have 8 00:00:40,150 --> 00:00:46,770 done so and found H of x no problem with integrals will still be provided with g of x. 9 00:00:46,860 --> 00:00:49,960 But this time we'll be asked to calculate f of x. 10 00:00:50,040 --> 00:00:51,570 It's antiderivative. 11 00:00:51,900 --> 00:00:57,450 Basically we're asking ourselves what function what I have had to differentiate to get g of x. 12 00:00:57,780 --> 00:01:01,240 We don't know how to do that yet but we're going to learn right now. 13 00:01:04,820 --> 00:01:09,060 There are two ways to calculate an antiderivative and find the area underneath the curve. 14 00:01:09,440 --> 00:01:16,370 Estimate the area using basic geometric area calculations or find the exact area using integrals. 15 00:01:16,490 --> 00:01:21,590 If you're taking a standard introductory calculus class you'll probably learn how to take an area estimation 16 00:01:21,710 --> 00:01:22,940 using Riemann sums. 17 00:01:22,940 --> 00:01:29,750 Trapezoidal rule where Simpson's approximation all of these methods are variations on the same theme. 18 00:01:29,960 --> 00:01:35,570 They use rectangles or trapezoids to follow the rough outline of the curve and reduce the area calculation 19 00:01:35,660 --> 00:01:37,940 down to basic geometric formulas. 20 00:01:38,240 --> 00:01:43,700 As you can imagine the more rectangles trapezoids you can use the more accurate your area estimation 21 00:01:43,700 --> 00:01:44,310 can be. 22 00:01:45,410 --> 00:01:50,450 If you imagine that you start using a larger and larger number of rectangles or trap asteroids until 23 00:01:50,450 --> 00:01:52,770 eventually use an infinite number. 24 00:01:52,820 --> 00:01:58,600 This is the point at which you're making an exact calculation using this infinite rectangle method to 25 00:01:58,600 --> 00:01:59,530 calculate area. 26 00:01:59,840 --> 00:02:04,030 Is similar to using the definition of the derivative to find the rate of change of a function. 27 00:02:04,950 --> 00:02:10,590 Both methods will get you the right answer but they're both basic and tedious in the same way that we 28 00:02:10,590 --> 00:02:15,210 learned better tools to calculate derivatives like product quotient and chain rule. 29 00:02:15,300 --> 00:02:17,620 We'll learn better ways to calculate integrals. 30 00:02:17,910 --> 00:02:20,080 Let's start with basic integral notation. 31 00:02:21,530 --> 00:02:26,190 Take this basic polynomial function as an example to take its antiderivative. 32 00:02:26,270 --> 00:02:32,480 We'll wrap it inside an integral and ADX the integral symbol basically says take the integral of this 33 00:02:32,480 --> 00:02:37,190 function and the DX says with respect to x. 34 00:02:37,250 --> 00:02:42,800 Think about both pieces of notation as a set they always have to be together when we're taking antiderivatives 35 00:02:43,520 --> 00:02:47,110 the integral symbol starts it and the DX finishes it. 36 00:02:47,290 --> 00:02:52,130 They're like bookends that tell us to take the integral of what comes in between them in the same way 37 00:02:52,130 --> 00:02:56,420 that you're able to take the derivative of a polynomial function by paying attention to one term at 38 00:02:56,420 --> 00:02:57,500 a time. 39 00:02:57,560 --> 00:03:02,320 We can take this function antiderivative by paying attention to one term at a time. 40 00:03:02,660 --> 00:03:08,990 To simplify will separate each term into its own integral when we deal with integrals we can put constant 41 00:03:09,050 --> 00:03:13,270 out in front of the integral symbol to further simplify the integration. 42 00:03:13,280 --> 00:03:18,470 Now we need to do is the opposite of what we've done in the past with derivatives. 43 00:03:18,580 --> 00:03:23,760 Let's focus and execute for a second to take the derivative of x cubed. 44 00:03:23,810 --> 00:03:31,240 We bring the three down in front and then subtract one from the exponent to get a result of 3 x squared. 45 00:03:31,320 --> 00:03:34,910 Remember that we want to reverse this process in order to take the integral. 46 00:03:35,250 --> 00:03:39,660 Well let's do that instead of subtracting one from the exponent. 47 00:03:39,660 --> 00:03:45,750 We'll add one then instead of multiplying the coefficient by the exponent we'll divide the coefficient 48 00:03:45,750 --> 00:03:47,290 by the exponent. 49 00:03:47,550 --> 00:03:52,550 As you can see we get an integral of one fourth X to the four. 50 00:03:52,630 --> 00:03:54,070 Do we really do that right. 51 00:03:54,350 --> 00:04:00,380 Well we can always check ourselves by taking the derivative or answer the derivative of one fourth X 52 00:04:00,380 --> 00:04:02,570 to the fourth is x cubed. 53 00:04:02,570 --> 00:04:03,840 Our original term. 54 00:04:03,950 --> 00:04:06,560 So we know we took the integral correctly. 55 00:04:07,070 --> 00:04:12,230 We can follow this pattern with the rest of the terms in our polynomial function will always add 1 to 56 00:04:12,230 --> 00:04:16,580 the exponent then divide the coefficient by the new exponent. 57 00:04:20,300 --> 00:04:23,710 Now is a good time to say a quick word about constants. 58 00:04:23,900 --> 00:04:27,090 Remember how constants would disappear when we took their derivatives. 59 00:04:27,480 --> 00:04:34,640 Well imagine what will happen if we're starting with G prime of X and trying to find g of X the antiderivative. 60 00:04:34,640 --> 00:04:39,330 How would we ever know that the plus 3 was part of the original function. 61 00:04:39,440 --> 00:04:44,690 We wouldn't nor we know the value of the constant assuming there even was one. 62 00:04:45,070 --> 00:04:50,180 The way that we account for this is by adding the constant of integration to the antiderivative that 63 00:04:50,180 --> 00:04:50,970 we calculate. 64 00:04:52,150 --> 00:04:55,530 We use a generic plus C to denote it. 65 00:04:55,630 --> 00:04:59,590 Remember that the constant of integration must always be added to your integral function. 66 00:04:59,590 --> 00:05:04,270 When you're dealing with indefinite integrals let's return to the integral we were looking at before 67 00:05:04,630 --> 00:05:06,240 and give our real final answer. 68 00:05:06,250 --> 00:05:10,240 But adding the constant integration to the end of our integrated function. 7748