[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.52,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.52,0:00:04.05,Default,,0000,0000,0000,,What I want to do in this video\Nis find the area of this region
Dialogue: 0,0:00:04.05,0:00:07.20,Default,,0000,0000,0000,,that I'm shading in yellow.
Dialogue: 0,0:00:07.20,0:00:11.14,Default,,0000,0000,0000,,And what might seem challenging\Nis that throughout this region,
Dialogue: 0,0:00:11.14,0:00:12.81,Default,,0000,0000,0000,,I have the same lower function.
Dialogue: 0,0:00:12.81,0:00:14.54,Default,,0000,0000,0000,,Or I guess the\Nlower boundary is y
Dialogue: 0,0:00:14.54,0:00:16.87,Default,,0000,0000,0000,,is equal to x squared\Nover 4 minus 1.
Dialogue: 0,0:00:16.87,0:00:19.24,Default,,0000,0000,0000,,But I have a different\Nupper boundary.
Dialogue: 0,0:00:19.24,0:00:20.70,Default,,0000,0000,0000,,And the way that\Nwe can tackle this
Dialogue: 0,0:00:20.70,0:00:23.29,Default,,0000,0000,0000,,is by dividing this\Narea into two sections,
Dialogue: 0,0:00:23.29,0:00:26.64,Default,,0000,0000,0000,,or dividing this region into two\Nregions, the region on the left
Dialogue: 0,0:00:26.64,0:00:28.25,Default,,0000,0000,0000,,and the region on\Nthe right, where
Dialogue: 0,0:00:28.25,0:00:30.73,Default,,0000,0000,0000,,for this first region,\Nwhich I'll do--
Dialogue: 0,0:00:30.73,0:00:34.31,Default,,0000,0000,0000,,I'll color even more in\Nyellow-- for this first region,
Dialogue: 0,0:00:34.31,0:00:35.95,Default,,0000,0000,0000,,over that entire interval in x.
Dialogue: 0,0:00:35.95,0:00:40.36,Default,,0000,0000,0000,,And it looks like x is\Ngoing between 0 and 1.
Dialogue: 0,0:00:40.36,0:00:44.28,Default,,0000,0000,0000,,y equals-- when x is equal to\N1, this function is equal to 1.
Dialogue: 0,0:00:44.28,0:00:47.22,Default,,0000,0000,0000,,When x is equal to 1, this\Nfunction is also equal to 1.
Dialogue: 0,0:00:47.22,0:00:48.81,Default,,0000,0000,0000,,So this is the point 1 comma 1.
Dialogue: 0,0:00:48.81,0:00:50.32,Default,,0000,0000,0000,,That's where they intersect.
Dialogue: 0,0:00:50.32,0:00:53.25,Default,,0000,0000,0000,,So for this section, this\Nsubregion right over here,
Dialogue: 0,0:00:53.25,0:00:57.42,Default,,0000,0000,0000,,y equals square root of x is the\Nupper function the entire time.
Dialogue: 0,0:00:57.42,0:00:59.23,Default,,0000,0000,0000,,And then we can have\Na-- we can set up
Dialogue: 0,0:00:59.23,0:01:02.86,Default,,0000,0000,0000,,a different-- we can\Nseparately tackle figuring out
Dialogue: 0,0:01:02.86,0:01:04.92,Default,,0000,0000,0000,,the area of this region.
Dialogue: 0,0:01:04.92,0:01:07.96,Default,,0000,0000,0000,,From x is equal to 1\Nto x is equal to 2,
Dialogue: 0,0:01:07.96,0:01:10.89,Default,,0000,0000,0000,,where y equals 2 minus\Nx, is the upper function.
Dialogue: 0,0:01:10.89,0:01:12.45,Default,,0000,0000,0000,,So let's do it.
Dialogue: 0,0:01:12.45,0:01:14.71,Default,,0000,0000,0000,,So let's first think\Nabout this first region.
Dialogue: 0,0:01:14.71,0:01:17.05,Default,,0000,0000,0000,,Well, that's going to be\Nthe definite integral from x
Dialogue: 0,0:01:17.05,0:01:19.64,Default,,0000,0000,0000,,is equal to 0 to\Nx is equal to 1.
Dialogue: 0,0:01:19.64,0:01:25.12,Default,,0000,0000,0000,,And our upper function is square\Nroot of x, so square root of x.
Dialogue: 0,0:01:25.12,0:01:28.39,Default,,0000,0000,0000,,And then from that, we want to\Nsubtract our lower function--
Dialogue: 0,0:01:28.39,0:01:32.32,Default,,0000,0000,0000,,square root of x minus x\Nsquared over 4 minus 1.
Dialogue: 0,0:01:32.32,0:01:39.20,Default,,0000,0000,0000,,
Dialogue: 0,0:01:39.20,0:01:42.40,Default,,0000,0000,0000,,And then of course,\Nwe have our dx.
Dialogue: 0,0:01:42.40,0:01:46.35,Default,,0000,0000,0000,,So this right over here, this is\Ndescribing the area in yellow.
Dialogue: 0,0:01:46.35,0:01:49.73,Default,,0000,0000,0000,,And you could imagine it, that\Nthis part right over here,
Dialogue: 0,0:01:49.73,0:01:51.66,Default,,0000,0000,0000,,the difference between\Nthese two functions
Dialogue: 0,0:01:51.66,0:01:53.16,Default,,0000,0000,0000,,is essentially this height.
Dialogue: 0,0:01:53.16,0:01:54.58,Default,,0000,0000,0000,,Let me do it in a\Ndifferent color.
Dialogue: 0,0:01:54.58,0:01:57.82,Default,,0000,0000,0000,,
Dialogue: 0,0:01:57.82,0:01:59.68,Default,,0000,0000,0000,,And then you\Nmultiply it times dx.
Dialogue: 0,0:01:59.68,0:02:03.39,Default,,0000,0000,0000,,You get a little\Nrectangle with width dx.
Dialogue: 0,0:02:03.39,0:02:06.60,Default,,0000,0000,0000,,And then you do that for each x.
Dialogue: 0,0:02:06.60,0:02:08.86,Default,,0000,0000,0000,,Each x you get a\Ndifferent rectangle.
Dialogue: 0,0:02:08.86,0:02:10.65,Default,,0000,0000,0000,,And then you sum them all up.
Dialogue: 0,0:02:10.65,0:02:14.57,Default,,0000,0000,0000,,And you take the limit as\Nyour change in x approaches 0.
Dialogue: 0,0:02:14.57,0:02:16.66,Default,,0000,0000,0000,,So as you get ultra,\Nultra thin rectangles,
Dialogue: 0,0:02:16.66,0:02:18.33,Default,,0000,0000,0000,,and you have an\Ninfinite number of them.
Dialogue: 0,0:02:18.33,0:02:21.06,Default,,0000,0000,0000,,And that's our definition,\Nor the Riemann definition
Dialogue: 0,0:02:21.06,0:02:22.82,Default,,0000,0000,0000,,of what a definite integral is.
Dialogue: 0,0:02:22.82,0:02:25.37,Default,,0000,0000,0000,,And so this is the area\Nof the left region.
Dialogue: 0,0:02:25.37,0:02:27.37,Default,,0000,0000,0000,,And by the exact same\Nlogic, we could figure out
Dialogue: 0,0:02:27.37,0:02:28.97,Default,,0000,0000,0000,,the area of the right region.
Dialogue: 0,0:02:28.97,0:02:30.68,Default,,0000,0000,0000,,The right region--\Nand then we could just
Dialogue: 0,0:02:30.68,0:02:32.13,Default,,0000,0000,0000,,sum the two things together.
Dialogue: 0,0:02:32.13,0:02:34.21,Default,,0000,0000,0000,,The right region, we're\Ngoing from x is equal to 0
Dialogue: 0,0:02:34.21,0:02:38.53,Default,,0000,0000,0000,,to x-- sorry, x is equal to\N1 to x is equal to 2, 1 to 2.
Dialogue: 0,0:02:38.53,0:02:42.13,Default,,0000,0000,0000,,The upper function is 2 minus x.
Dialogue: 0,0:02:42.13,0:02:47.22,Default,,0000,0000,0000,,And from that, we're going to\Nsubtract the lower function,
Dialogue: 0,0:02:47.22,0:02:49.66,Default,,0000,0000,0000,,which is x squared\Nover 4 minus 1.
Dialogue: 0,0:02:49.66,0:02:53.78,Default,,0000,0000,0000,,
Dialogue: 0,0:02:53.78,0:02:56.06,Default,,0000,0000,0000,,And now we just\Nhave to evaluate.
Dialogue: 0,0:02:56.06,0:02:58.80,Default,,0000,0000,0000,,So let's first simplify\Nthis right over here.
Dialogue: 0,0:02:58.80,0:03:02.10,Default,,0000,0000,0000,,This is equal to the\Ndefinite integral
Dialogue: 0,0:03:02.10,0:03:09.22,Default,,0000,0000,0000,,from 0 to 1 of square root of x\Nminus x squared over 4 plus 1,
Dialogue: 0,0:03:09.22,0:03:12.02,Default,,0000,0000,0000,,dx-- I'm going to write\Nit all in one color now--
Dialogue: 0,0:03:12.02,0:03:18.71,Default,,0000,0000,0000,,plus the definite integral\Nfrom 1 to 2 of 2 minus x,
Dialogue: 0,0:03:18.71,0:03:21.33,Default,,0000,0000,0000,,minus x squared over 4.
Dialogue: 0,0:03:21.33,0:03:25.33,Default,,0000,0000,0000,,Then subtracting a negative is\Na positive 3-- or a positive 1.
Dialogue: 0,0:03:25.33,0:03:26.65,Default,,0000,0000,0000,,We could just add it to this 2.
Dialogue: 0,0:03:26.65,0:03:29.33,Default,,0000,0000,0000,,And so this 2 just becomes a 3.
Dialogue: 0,0:03:29.33,0:03:34.75,Default,,0000,0000,0000,,I said 2 minus\Nnegative 1 is 3, dx.
Dialogue: 0,0:03:34.75,0:03:36.70,Default,,0000,0000,0000,,And now we just have to\Ntake the antiderivative
Dialogue: 0,0:03:36.70,0:03:39.31,Default,,0000,0000,0000,,and evaluate it at 1 and 0.
Dialogue: 0,0:03:39.31,0:03:42.13,Default,,0000,0000,0000,,So the antiderivative\Nof this is-- well,
Dialogue: 0,0:03:42.13,0:03:43.48,Default,,0000,0000,0000,,this is x to the 1/2.
Dialogue: 0,0:03:43.48,0:03:44.73,Default,,0000,0000,0000,,Increment it by 1.
Dialogue: 0,0:03:44.73,0:03:47.50,Default,,0000,0000,0000,,Increment the power by\N1, you get x to the 3/2,
Dialogue: 0,0:03:47.50,0:03:49.20,Default,,0000,0000,0000,,and then multiply\Nby the reciprocal
Dialogue: 0,0:03:49.20,0:03:53.65,Default,,0000,0000,0000,,of the new exponent-- so\Nit's 2/3 x to the 3/2.
Dialogue: 0,0:03:53.65,0:03:56.41,Default,,0000,0000,0000,,Minus-- the antiderivative\Nof x squared over 4
Dialogue: 0,0:03:56.41,0:04:02.16,Default,,0000,0000,0000,,is x to the third, divided by 3,\Ndivided by 4, so divided by 12,
Dialogue: 0,0:04:02.16,0:04:03.66,Default,,0000,0000,0000,,plus x.
Dialogue: 0,0:04:03.66,0:04:05.51,Default,,0000,0000,0000,,That's the antiderivative of 1.
Dialogue: 0,0:04:05.51,0:04:09.59,Default,,0000,0000,0000,,We're going to\Nevaluate it at 1 and 0.
Dialogue: 0,0:04:09.59,0:04:11.64,Default,,0000,0000,0000,,And then here the\Nantiderivative is
Dialogue: 0,0:04:11.64,0:04:19.67,Default,,0000,0000,0000,,going to be 3x minus x\Nsquared over 2 minus x
Dialogue: 0,0:04:19.67,0:04:22.03,Default,,0000,0000,0000,,to the third over 12.
Dialogue: 0,0:04:22.03,0:04:24.45,Default,,0000,0000,0000,,Once again, evaluate it\Nat-- or not once again.
Dialogue: 0,0:04:24.45,0:04:28.46,Default,,0000,0000,0000,,Now we're going to\Nevaluate at 2 and 1.
Dialogue: 0,0:04:28.46,0:04:30.61,Default,,0000,0000,0000,,So over here, you evaluate\Nall of this stuff at 1.
Dialogue: 0,0:04:30.61,0:04:35.69,Default,,0000,0000,0000,,You get 2/3 minus 1/12 plus 1.
Dialogue: 0,0:04:35.69,0:04:38.41,Default,,0000,0000,0000,,And then from that, you\Nsubtract this evaluated at 0.
Dialogue: 0,0:04:38.41,0:04:41.01,Default,,0000,0000,0000,,But this is just all\N0, so you get nothing.
Dialogue: 0,0:04:41.01,0:04:44.26,Default,,0000,0000,0000,,So this is what the yellow\Nstuff simplified to.
Dialogue: 0,0:04:44.26,0:04:46.74,Default,,0000,0000,0000,,And then this purple stuff,\Nor this magenta stuff,
Dialogue: 0,0:04:46.74,0:04:51.07,Default,,0000,0000,0000,,or mauve, or whatever color this\Nis, first you evaluate it at 2.
Dialogue: 0,0:04:51.07,0:04:58.33,Default,,0000,0000,0000,,You get 6 minus-- let's see,\N2 squared over 2 is 2, minus 8
Dialogue: 0,0:04:58.33,0:04:58.92,Default,,0000,0000,0000,,over 12.
Dialogue: 0,0:04:58.92,0:05:01.52,Default,,0000,0000,0000,,
Dialogue: 0,0:05:01.52,0:05:03.65,Default,,0000,0000,0000,,And then from that,\Nyou're going to subtract
Dialogue: 0,0:05:03.65,0:05:05.46,Default,,0000,0000,0000,,this evaluated at 1.
Dialogue: 0,0:05:05.46,0:05:13.27,Default,,0000,0000,0000,,So it's going to be 3 times 1--\Nthat's 3-- minus 1/2 minus 1
Dialogue: 0,0:05:13.27,0:05:14.60,Default,,0000,0000,0000,,over 12.
Dialogue: 0,0:05:14.60,0:05:16.27,Default,,0000,0000,0000,,And now what we're\Nessentially left with
Dialogue: 0,0:05:16.27,0:05:17.89,Default,,0000,0000,0000,,is adding a bunch of fractions.
Dialogue: 0,0:05:17.89,0:05:19.18,Default,,0000,0000,0000,,So let's see if we can do that.
Dialogue: 0,0:05:19.18,0:05:20.93,Default,,0000,0000,0000,,It looks like 12 would\Nbe the most obvious
Dialogue: 0,0:05:20.93,0:05:22.30,Default,,0000,0000,0000,,common denominator.
Dialogue: 0,0:05:22.30,0:05:29.43,Default,,0000,0000,0000,,So here you have 8/12\Nminus 1/12 plus 12/12.
Dialogue: 0,0:05:29.43,0:05:31.31,Default,,0000,0000,0000,,So this simplifies\Nto-- what's this?
Dialogue: 0,0:05:31.31,0:05:36.44,Default,,0000,0000,0000,,This is 19/12, the part\Nthat we have in yellow.
Dialogue: 0,0:05:36.44,0:05:40.19,Default,,0000,0000,0000,,And then this business,\Nlet me do it in this color.
Dialogue: 0,0:05:40.19,0:05:43.28,Default,,0000,0000,0000,,So 6 minus 2, this is\Njust going to be 4.
Dialogue: 0,0:05:43.28,0:05:51.10,Default,,0000,0000,0000,,So we can write this as\N48/12-- that's 4-- minus 8/12.
Dialogue: 0,0:05:51.10,0:05:54.46,Default,,0000,0000,0000,,And then we're going to have to\Nsubtract a 3, which is 36/12.
Dialogue: 0,0:05:54.46,0:05:57.17,Default,,0000,0000,0000,,
Dialogue: 0,0:05:57.17,0:06:02.41,Default,,0000,0000,0000,,Then we're going to add to\N1/2, which is just plus 6/12,
Dialogue: 0,0:06:02.41,0:06:06.03,Default,,0000,0000,0000,,and then we're\Ngoing to add a 1/12.
Dialogue: 0,0:06:06.03,0:06:10.73,Default,,0000,0000,0000,,So this is all going to simplify\Nto-- let's see, 48 minus 8
Dialogue: 0,0:06:10.73,0:06:18.41,Default,,0000,0000,0000,,is 40, minus 36 is 4, plus\N6 is 10, plus 1 is 11.
Dialogue: 0,0:06:18.41,0:06:21.61,Default,,0000,0000,0000,,So this becomes plus 11/12.
Dialogue: 0,0:06:21.61,0:06:23.03,Default,,0000,0000,0000,,Let me make sure\NI did that right.
Dialogue: 0,0:06:23.03,0:06:28.83,Default,,0000,0000,0000,,48 minus 8 is 40,\Nminus 36 is 4, 10, 11.
Dialogue: 0,0:06:28.83,0:06:30.04,Default,,0000,0000,0000,,So that looks right.
Dialogue: 0,0:06:30.04,0:06:31.96,Default,,0000,0000,0000,,And then we're ready\Nto add these two.
Dialogue: 0,0:06:31.96,0:06:36.01,Default,,0000,0000,0000,,19 plus 11 is equal to 30/12.
Dialogue: 0,0:06:36.01,0:06:38.29,Default,,0000,0000,0000,,Or if we want to simplify\Nthis a little bit,
Dialogue: 0,0:06:38.29,0:06:40.99,Default,,0000,0000,0000,,we can divide the numerator\Nand the denominator by 6.
Dialogue: 0,0:06:40.99,0:06:44.96,Default,,0000,0000,0000,,This is equal to\N5/2, or 2 and 1/2.
Dialogue: 0,0:06:44.96,0:06:45.63,Default,,0000,0000,0000,,And we're done.
Dialogue: 0,0:06:45.63,0:06:50.56,Default,,0000,0000,0000,,We figured out the area\Nof this entire region.
Dialogue: 0,0:06:50.56,0:06:53.26,Default,,0000,0000,0000,,It is 2 and 1/2.
Dialogue: 0,0:06:53.26,0:06:53.76,Default,,0000,0000,0000,,