00:00Hi everybody. So in this
00:02lesson we're going to look
00:03at the trapezoidal rule. Now
00:05what the trapezoidal rule does
00:07is it gives us an
00:08estimate for the area under
00:10the curve. Now we already
00:12did in a previous lesson.
00:13We did the integral. We
00:15said the integral which is
00:16this from A to B
00:17of y dx gives you
00:19the exact area and we
00:20agreed that that was pretty
00:22incredible. But the integral from
00:24here to here gives me
00:27an exact
00:28area underneath the curve. Right,
00:30and when we did this,
00:31we actually got an answer
00:32of 26. Let's just keep
00:35that in mind when we
00:36do the trapezoidal rule. Because
00:38we're not going to get
00:3926, but hopefully we'll get
00:40close to it. Now, what
00:43it does is, it splits,
00:45we can split this up
00:46into a number of different
00:49trapezoids. Now, what is a
00:50trapezoid? Well, it's this shape
00:54that you probably remember.
00:56which has a pair of
00:58parallel sides. Now there's some
01:01debate as to whether it
01:03has exactly one pair of
01:05parallel sides or at least
01:07one pair of parallel sides,
01:10but it doesn't really matter
01:12in our situation. And this
01:14is the height of the
01:18trapezoid. So what we do
01:20is, imagine we split this
01:21into two different trapezoids.
01:24We did one, two. We
01:28got the area of this,
01:30and we got the area
01:30of this, and we added.
01:32So that would give me
01:32an approximation of the area
01:34under the curve. It wouldn't
01:35be a very good one,
01:36because obviously I've added on
01:37this extra bit of area
01:39here that I didn't want.
01:40But what if I, that's
01:42just two trapezoids. What if
01:44I did, let's say what
01:46if I did six trapezoids,
01:48one, two, three,
01:52four, five, six, and join
01:57these. Well, now you're starting
01:59to see this is a
02:01much better approximation of the
02:04area of the curve. Now
02:05if I did 100 chopper
02:06solids, that would be a
02:08really, really good approximation of
02:10the area of the curve.
02:12So what we're gonna do
02:14is, well, let's do this
02:15example, and we'll see how
02:17close to 26 we actually
02:19get.
02:20Now the way this formula
02:21works is, well, one you're
02:24given it in the form
02:24of a booklets, section 5
02:26.8. And what it does
02:27is the area of a
02:30trapezoid, a trapezium, I think
02:33I grew up calling these
02:34things. The area is a
02:39half of x plus y,
02:41because it gets the average
02:42length of those two, the
02:44two parallel size, and then
02:46times it by h. So
02:48what this
02:48formula does is it gets,
02:50well it basically sums up
02:52all those different trapezoids and
02:53it's pretty, it'll be pretty
02:54easy for you to do.
02:55I'm not going to derive
02:56it myself but you could
02:57even do it, you could
02:58do it if you want
02:59it. Just, you just kind
03:00of get all the trapezoids,
03:02lengths and, and get the
03:04height and then rearrange it
03:06and you'll get it into
03:07this form. It's not that
03:08difficult. But the height is,
03:12well the height, because it's
03:14between the distance between the
03:16parallel size, our height is
03:17actually going to be going
03:18this way. So if I
03:20did two trapezoids, my height
03:23would actually be three. It
03:24would be this height here.
03:26So the way we get
03:27the height, the H, is
03:29we first see how many
03:31trapezoids does he want. So
03:33this question is used to
03:34trapezoidal rule and four even
03:36intervals to find an approximation
03:38for there under the curve
03:39in the interval two to
03:40eight. So H is B
03:43minus A over
03:44Now B and A are
03:47the limits, so we're going
03:48from 2 to 8. This
03:50is 2, this is 8.
03:51So from 2 to 8,
03:55the, from 2 to 8,
03:57this would be 8 minus
03:592 over n, which is
04:034, because he wants 4,
04:064 even intervals or 4
04:07trapezoids. 6 over 4, which
04:09is 1 .5. So the,
04:12the high
04:12height of our trapezoids, which
04:14is actually going to be
04:15our length here, is 1
04:16.5. So I'm going to
04:17go 1 .5 there. 1
04:20.5 is there. 1 .5
04:23is there. And you can
04:25see why it's 1 .5
04:27because that's a, that is
04:30for evenly spaced trapezoids. Now,
04:37it's worth thinking, because I
04:39said the integral is 26.
04:40Air is exactly 26, but
04:42when I do this trapezoidal
04:44rule, am I gonna get
04:46more than 26 or less
04:47than 26? Well, I'm actually
04:49gonna get more than 26,
04:50because each of these has
04:51a tiny, look, it has
04:52a tiny little bit more
04:54there that we're gonna be
04:56adding that isn't actually below
04:57the curve. So I'm gonna
04:58get probably 26 points something.
05:00I don't think I'm gonna
05:01get 27, because it's not
05:03that big. There's not that
05:04big a gap. Okay, so
05:06I have H, great. No.
05:08I'm going to say the
05:10integral from 2 to 8
05:13of y dx is approximately
05:16equal to, I'm going to
05:17write out this formula. It's
05:19a half h y0 plus
05:23yn plus 2 times y1
05:27plus y2 plus dot dot
05:30dot. Let's just leave it
05:31at that. Okay, so I
05:33have, right, I have x
05:36z
05:36I have x1, this is
05:40actually x2, this is x3,
05:45and this is x4. But
05:47he wants y0, y1, y2,
05:50y3, and y4, because as
05:54I say this, to do
05:54with this rule here, he
05:55wants this length and this
05:57length and this length and
05:59this length. And we're gonna
06:00do this thing. And you'll
06:02see actually that's why he
06:03only has y0 and y4.
06:04n once, because this goes
06:07for this trapezoid, but y1
06:10and y2, the middle ones,
06:13he adds twice, because this,
06:14like x1 will go with
06:16this trapezoid and this trapezoid,
06:18that's where this two of
06:19them. Anyway, I'm not going
06:21to get into too much
06:22detail about the formula, as
06:23I say, it's fairly straightforward.
06:26Now, what I want to
06:27do is get, okay, I
06:30have x and I have
06:33Y. So I'm actually going
06:35to make a little table
06:36here. I'm going to make
06:38a little table. So when
06:41X is, this is two,
06:44what is Y? Then when
06:46X is 3 .5, what
06:49is Y? Then when X
06:50is 5, what is Y?
06:526 .5, and then 8.
06:55This is my X0, X1,
06:57X2, X3, X4. So this
06:59will be my Y0, Y1.
07:01y2, y3, y4. OK, so
07:04when I have x, how
07:06do I find y? Would
07:07I just sub it into
07:09the function or whatever the
07:14formula is here? So I
07:15have y is a third
07:15x squared minus 3x plus
07:1610. Now you could, no
07:19problem, sub it into each
07:20one of them and get
07:21your answer. I wanted to
07:22show you a little trick
07:24for, I just want to
07:27show you a little trick.
07:29to do this pretty quickly.
07:31So we have two, I've
07:34opened up a spreadsheet here.
07:36So I have two, three,
07:38point five, five, six, point
07:43five, and eight. Now, what
07:48I want is a third
07:49x squared minus three x
07:50plus 10. So I'm gonna
07:53do a third,
07:57And then instead of doing
08:00x squared, I'm going to
08:01click this. Yeah, that's not
08:03what I want you to
08:04do. I'm going to do,
08:06let me do that again.
08:09I'm going to do a
08:10third. And then I'm going
08:12to do times, I forgot
08:13the times, which is important.
08:14I'm going to a third
08:15times, and now instead of
08:19writing in two squared or
08:22whatever, I'm actually going to,
08:23the way you would an
08:24Excel,
08:25click this and it gives
08:26me a1. So look at
08:29that. That's going to give
08:30me a1 squared. Now if
08:33this freaks you out or
08:34if you really don't like
08:35to look at this, no
08:35problem, just sub in each
08:37one. It'll take you maybe
08:39two minutes x for this
08:40is to kind of save
08:41you time. If you like
08:42this kind of thing. So
08:43that's going to be a
08:44third x squared or third
08:45a1 squared minus three times
08:49a1 again. And then
08:53and plus 10, press enter.
09:02I'm sorry, I feel like
09:06an idiot here, right? I
09:07should have pressed equals. So
09:09in Excel, you would press
09:11equals that. So press enter
09:13and now I get 16
09:14over three. Now what I
09:16can do is, in the
09:19corner here, pull this down.
09:21And if you're good with
09:22Excel, you'll be happy with
09:25this. So when you pull
09:26it down like this, you
09:29get it fills in all
09:32those answers for you. So
09:33I've done them quite quickly,
09:35rather than subbing it in
09:39each time. Again, if you
09:40don't like this, no problem.
09:44Just sub them in individually.
09:46And that's fine. So I
09:48have 16
09:49over 3, 16 over 3,
09:553 .5, 8, 3, 3,
09:573, 3, 3, 3, 10
10:03over 3, 10 over 3,
10:074 .58333 and 22 over
10:173, 22 over 3. Okay,
10:21so these are my, this
10:23is my y0, this is
10:25my y4, and these are
10:28the ones in between. So
10:29the area, so the area
10:32is approximately equal to a
10:35half times H, what is
10:38H, 1 .5 times big
10:42brackets. And I do 16
10:44over three,
10:45Plus, let me write that
10:48properly actually, I do 16
10:50over 3 plus y0, the
10:54first y plus the last
10:55one plus 22 over 3.
10:59Close bracket plus 2 times,
11:01and then these ones, 3
11:04.5833 plus 10 over 3
11:09plus 4 .5833.
11:13three, close this bracket and
11:19close this bigger bracket there.
11:23So this is gonna give
11:25me, this is my trapezoidal
11:27rule and it's gonna give
11:28me the area under the
11:29curve. It'll give me, I'm
11:32certain unless I've made a
11:34total mess of this, it
11:35is gonna give me an
11:37answer that's 16 points something,
11:40sorry, 26
11:41point something. So let's write
11:43it in exactly how I
11:47see it. So it's 1,
11:501 over 2 times 1
11:55.5, 2 brackets. Now I
11:58have, let's just do 16
12:00over 3 like this, 16
12:02over 3 plus 22 over
12:053, close that bracket plus
12:082 times
12:093 .5, 8, 3, 3,
12:133, plus 10 over 3,
12:18plus 4 .5, 8, 3,
12:213, 3, close this bracket,
12:23close that bracket, press enter
12:2526 .75 perfect, that's what
12:28I was expecting, 26 .75.
12:33So when it says use
12:35the trapezoidal rule, four even
12:37intervals, one to
12:37three, four, to find an
12:39approximation for the area under
12:40the curve in the interval
12:41two to eight. That is
12:44my approximation of the area.
12:48Okay, so a few things.
12:50What, what do you think
12:52of this question? Well, if
12:55I was a student, I
12:58would, I would actually think,
13:00right, that's, that looks really,
13:02that looks really annoying and
13:04really difficult and there's lots
13:05of
13:06annoying things going on with
13:07the calculator. But it's not
13:09actually that difficult to question
13:11if you can just if
13:12you can decide what exactly
13:15your your x's and your
13:16y's are because you have
13:17this formula and the places
13:19you're going to go wrong
13:20is just putting the pressing
13:22the wrong button in the
13:23calculator. I actually think it's
13:26a nice question and when
13:27you see it you should
13:28think okay good I know
13:29how to do that question.
13:30Now the only way you'll
13:31know how to do it
13:31is obviously if you've practiced
13:33quite a number of
13:34these beforehand or throughout the
13:38year, the year, or the
13:39two years, or whatever it
13:39is. So yeah, you've a
13:43formula, you find H depending
13:45on what the question says.
13:47Then you get your X's,
13:48find your Y's if you
13:50want to do the fancy
13:50way that I did it
13:51fine. But if you're in
13:54the pressure of an exam
13:55and you really don't, you're
13:58really not comfortable with that,
13:59don't even bother. Just sub
14:00it, sub two into this.
14:02then sub 3 .5 into
14:04this, then sub 5 into
14:04this, then sub 6 .5
14:06into this, and then age
14:07into this, and you'll get
14:08these same values. Okay, hopefully
14:11that made sense. I'll talk
14:14to you in the next
14:14lesson.