Trapezium Rule: Example

00:01 So let's look at this example.

00:03 We are using the trapezium rule to approximate and this is important to understand because we're not accurately finding the area, we're just approximating it.

00:12 The area between one and zero, we've been given a function and it says five ordinates.

00:18 So let's write this out.

00:19 This is what we're doing.

00:20 So we're looking for the area between one and zero, 3x plus 1 to the power of 2 dx.

00:29 Now, all of this notation will make a lot more sense when we do integration.

00:33 But basically, this sign here just means find the area or the integral.

00:38 This is the integral sign.

00:40 These here are your limits.

00:42 So these are your X limits of zero and one and I'll sketch that in a second to show you.

00:47 This is obviously our function and this thing here just says, "with respect to X." So similar as the dx that we were using in differentiation with respect to X.

01:01 We'll formally define this shortly when we do integration the faster way.

01:06 So, first of all, the first fact that I can use, they've given us five ordinates, which to me means that we have four strips or four trapeziums.

01:18 Now, I don’t really know what this graph looks like but I'm just going to roughly give us an idea that we have some form of a function.

01:25 So, we are looking for the area between zero and one, so between there and there, for example, and we're splitting it up into trapeziums.

01:34 Now, we've got four trapeziums here.

01:36 So, if we make it better so it goes through zero.

01:40 Let's make it a function so it looks like that.

01:45 We've got four strips here.

01:48 So what we want to do is we want to split this into four trapeziums.

01:52 That means we're going to have five ordinates.

01:54 So we have zero, which is our first ordinate, and one as our last ordinate.

01:58 So we have zero, one, two, three, four, and five.

02:04 So we have one, two, three, four, and five.

02:06 We've got one extra ordinate.

02:08 Let's just fix that here.

02:11 So I should have five ordinates.

02:13 I'm just going to make sure that we're doing that correct.

02:15 So we've got one, two, three, four, and five.

02:21 Very thin strips but that's what integration really is.

02:23 We're trying to make them infinitesimally small here.

02:28 So, here, we have our trapeziums or our strips which means that we have five ordinates.

02:37 Okay.

02:38 Now, we even told that this goes from zero to one.

02:41 And the first thing that we need to do is calculate the height.

02:45 So we want to know the height of each trapezium, okay? So, if we know what 0 to 1 is, we need to know what the H is, so what are these little increments there, these little steps for each individual trapezium.

02:58 So we want to know what the height is.

03:01 Now, in order to do that, if you really think about it, how can you split this little length here between 1 and 0 into these number of trapeziums.

03:10 So, if I give you the formula for this, you can say that the height is B minus A over N.

03:18 B and A are your X limits.

03:21 So this is just something that you can learn.

03:23 These are your X limits and N is your number of strips.

03:30 So again, I'm sure you'd be able to figure this out just by looking at it, because you want to go between 0 and 1 and you want to split it into four trapeziums.

03:40 So you can calculate the height from here.

03:42 We can say the height is, this is your upper limit and this is your lower limit.

03:46 So you do one minus zero, B minus A, and the number of strips we said was four.

03:52 So the height of each trapezium is 1 over 4 or 0.25.

03:59 Okay.

04:00 Let's call this step one.

04:04 So, when you're using the trapezium rule, the first thing to do is calculate your height because we need that for the formula.

04:11 Step two, we now have to find all our Y values, because let me just remind you of the area.

04:16 The area says of a trapezium rule is approximately H over 2, Y0 plus YN, that's the last one, plus two times all the other, Y1, Y2, however many that you have.

04:29 So that's just the general formula.

04:33 So, we've calculated our H.

04:35 We now need to calculate all of these N values.

04:38 In order to calculate these N values, you obviously need all your X values.

04:44 So, if we make a little table and we start to say that we've got all our X values and our Y values, we're going to start with our first X value.

04:59 So we want to know what the Y value is here when we put in our first X value at zero.

05:06 And then we're going up by H steps for each one of them.

05:09 So we're taking a small, little step of H to go to the next X value.

05:14 So, if we go up in steps of 0.25, I've got zero.

05:19 The next ordinate will be 0.25.

05:22 The next ordinate will be 0.5.

05:24 The next ordinate will be 0.75.

05:27 And the last ordinate will be 1, which is what it says here to go between 0 and 1.

05:34 And you can double check, you can count.

05:36 You've got one, two, three, four, five ordinates, which is what they wanted.

05:43 To find your Y ordinates, you now need to substitute these values into your Y function.

05:48 So these here is your Y functions.

05:50 We're going to substitute each one of them into Y.

05:53 So we're going to have three 3 times 0 plus one squared, which in this case just gives me one.

06:01 The next one, we have 3 times 0.25 plus one squared, which will work out.

06:08 We've got 3 times 0.5 plus one squared that will need to work out.

06:14 Once again, you can treat them as fractions and it might be quite easy to add them up, which we can do in a minute.

06:19 We've got 3 times 0.75 plus one squared.

06:23 And then we've got 3 times 1, so brackets around this, plus one squared.

06:29 I shouldn't have the plus in the inside.

06:32 Now, for a trapezium rule, you should be allowed to use a calculator when we're using one, when we're using -- deriving the rule.

06:41 Up to here, we can do this without a calculator because we can just work within fractions.

06:46 So, we can say that that's just going to give us one.

06:49 This is the same as 3 times 1 over 4, plus one all squared, which gives you three over four plus one, which is essentially just 7 over 4 all squared.

07:04 So we're just dealing with fractions here.

07:06 This is the same as 3 times a half plus one, which is 3 over 2 plus 2, which will be 5 over 2 squared.

07:18 The next one, if I write this one out, so we've got three and we've got 0.75 which is 3 over 4 , it's three quarters plus one all squared.

07:29 If you work this out, that gives you 9 over 4, which when you add to 1 will give you 13 over 4 all squared.

07:38 And lastly, the last one should be fairly straightforward.

07:41 We've got 3 times 1 plus 1 which gives you 4 squared here which is just 16.

07:49 Okay So as you can tell that this will be a little bit easier when we use a calculator, but we can get away, we're not doing it if we really have to.

07:58 But at no point, with an exam question, I want you to use a trapezium rule especially of this level of complication without a calculator.

08:07 So, now that we know all our Y values, this is my first Y value.

08:12 Y0, this will be Y1, Y2.

08:15 Obviously, I haven't worked them out as numbers yet, Y3 and Y4 are also my YN, if you want to call it that.

08:23 Let's put it all into the trapezium rule now.

08:25 So we can say that the area is approximately H over two.

08:29 H is this value, so 0.25 over 2.

08:33 Take your first Y and your last Y.

08:36 So my first Y is 1, my last Y is 16 plus two times all the other Y values here.

08:43 So, we're going to have 7 over 4 squared plus 5 over 2 squared plus 13 over 4 squared.

08:54 So all I need to do is work this out.

08:56 Do this in a calculator.

08:58 If I just rewrite this as 0.25 over 2, 1 plus 16 gives me 17 plus 2.

09:05 I can square these.

09:06 Up to that point, I can do -- that's 49 over 16 plus 25 over 4 plus 169 over 16.

09:15 And if you just put that all into your calculator now, we end up with a final answer of approximately 7.0937.

09:25 And remember that if you were using a calculator, a lot of it becomes easier.

09:29 So there are a lot of little calculations that I've done that you wouldn't have to do because you can just put the numbers straight into the calculator and find the area.

09:38 And you can see that it's quite an exhaustive process.

09:40 If you have to do integration from scratch, if you have to do integration as an approximation, I hope that you started to appreciate that it's not the easiest of things to do, and even if it's easy, it's tedious.

09:52 It takes a long time, it takes a lot of effort and you definitely need some help.

09:56 Or, if you're doing it from scratch, it would just take much longer time.

10:01 So, what we'll do now is we'll move on to a faster method of integration which will make our lives a lot easier and hopefully you'll see us coming to the same result more accurately and more easily.