Basic Integration: Exercise 3

00:01 So the last question is asking us to use the trapezium rule with five ordinates.

00:06 Now, we're using the trapezium rule just in order to appreciate how good integration actually is, how much easier it makes our lives.

00:14 So we're going to practice the trapezium rule again.

00:17 Remember that it is a little bit tedious and you will need to use your calculator for this because we're dealing with square roots.

00:23 But we go through it step by step and hopefully at the end, we're fairly amazed with how the approximation is very similar to the actual answer.

00:31 So let's try this question out.

00:33 We have the integral between three and one, one over root x.

00:40 And it says that we are using five ordinates.

00:43 So remember what to do.

00:44 Firstly, just change your ordinates to strips or the number of N or trapeziums whatever you want to call them.

00:51 So five ordinates means four strips because the number of strips is always one less than the ordinates and this is also our N value.

01:01 Let's just go over the formula for trapezium rule.

01:04 So the trapezium formula stated that you have area H over 2, Y0 plus YN plus 2 times all the other Ys.

01:14 So however many you have, you can just put them in there.

01:19 So that means that we need to find everything in this formula step by step.

01:23 The first thing that we need to do is calculate our h.

01:27 And if you look back at our formula for h, we said H is the height of each trapezium, is b minus a over n.

01:36 B is the upper limit and A is the lower limit.

01:40 So we'll do 3 minus 1 over 4 to give me 2 over 4, which gives me an answer of a half or a 0.5.

01:49 If you prefer to work in decimals, you can use decimals.

01:53 If you like working in fractions, then that's also fine.

01:56 We are going to put all these numbers into the calculator so it doesn’t matter which version you use.

02:02 The next step, we need to calculate all of these Y values.

02:06 Now, in order to find those Y values, you firstly need to know all your Xs.

02:11 So, we're going to calculate each X starting from 1 going up in steps of 0.5.

02:18 So, if we are to imagine what's happening here on our graph.

02:22 So if we start with 1, our next ordinate will be, when you add 0.5 to it, so that will be 1.5.

02:30 Our next ordinate will be, when you add another 0.5 to it, so that will be 2.

02:34 And these will be all your X values until you get to 3.

02:39 And that's how you calculate your X ordinates.

02:42 So if we put it in a little table here to make it clearer, we have our X values here and our Y values.

02:49 Start with the lowest value of X.

02:52 So, imagine we have some sort of a curve and this is the area that you're approximating.

02:57 So start with the lowest value of X which is one.

03:01 Our next value will be 1.5.

03:04 Next value will be 2.

03:05 Next value will be 2.5.

03:07 And the last value will be 3.

03:09 And I know that that's my last value because that's as far as I need to go.

03:13 So as soon as you hit the top value, you stop.

03:16 Check that it's the correct number of ordinates.

03:19 So if it isn't, there's something wrong with your H.

03:23 So go back and calculate your H again.

03:25 But if it is, then you've done it correctly.

03:27 So five ordinates, one, two, three, four, five.

03:31 So I'm confident that my H values are correct because I've got five ordinates between 1 and 3.

03:39 The next thing I need to do is calculate my Y, and my Y comes from here.

03:44 So my Y function is 1 over root X.

03:47 I should have a dx at the end.

03:50 So my Y value is 1 over root X.

03:53 So I'm going to put X here to find each individual Y.

03:57 So the first one, I'll have 1 over root 1 to just give me 1.

04:02 The next one, I'll have 1 over root 1.5.

04:06 This gives me 0.8165 to 4 decimal places when I put it in my calculator.

04:13 The next one will be 1 over root 2.

04:17 This gives me 0.7071 to 4 decimal places.

04:22 The next one will be 1 over root 2.5, which according to me is 0.6325 to 4 decimal places.

04:31 And the last one is 1 over root 3, which you can leave like this if you want to stay more accurate or because we've converted everything into decimals.

04:40 That gives me 0.57735 or 34 -- I should do since I'm going up to four decimal places.

04:52 Okay. All my Y values have been calculated.

04:54 Remember, this is going to be Y0, Y1, Y2, Y3, and Y4.

05:00 So technically, this is my first and this is my last value and these are all the values in between.

05:07 So let's put it all into our formula to just get rid of this graph here, so we make some more space.

05:13 So when I put this into my formula, I just need to replace all my Y values with these numbers now.

05:19 So my area is approximately H is 0.5 over 2.

05:27 Take your first Y value, which is this one, and your last Y value.

05:32 So I have 1 plus 0.5774 plus two times all the other Y values here.

05:40 So we're now going to add all of them here in the bracket.

05:44 So 0.8165 plus, so do that here, 0.7071 plus 0.6325.

05:57 Put it all into your calculator.

05:58 Just be careful that you do this step by step so you don’t confuse your calculator.

06:02 This gives me an approximate area of 1.472 to 3 decimal places.

06:11 So that's the area that I've approximated, and you see, once again, that it's a fairly tedious method.

06:16 Now, if I do this quickly just using our standard integration method.

06:21 So if we just do that here on the side.

06:24 If we do integrate one over root X dx between the limits of 1 and 3, the first thing just as differentiation, I'm not happy with this being at the bottom.

06:36 Remember, root X has a power to the half.

06:39 So I can bring that up.

06:41 So I can rewrite this as X to the minus a half.

06:44 I haven't integrated.

06:45 I just brought it up, dx.

06:48 So remember what we've done root X is the same as X to the power of a half.

06:53 If you brought it up from the denominator, it will change sign.

06:57 So it was positive as a denominator, when you bring it up, it becomes minus.

07:02 When you integrate, add 1 to the power divide by a new power.

07:06 So minus a half, let's just write that here, minus a half plus one, that gives you positive a half, so divided by a new power.

07:17 I don’t need to put C at the end because this is a definite integral, so I'm going to put numbers in.

07:24 When you have a fraction being divided, a number being divided by a fraction, this bottom number can just come to the top.

07:30 So we can rewrite this as 2x to the half.

07:34 That half comes from here minus a half plus 1.

07:37 And then I have to put my limits in of 1 and 3.

07:42 So, this is the same as 2 root X.

07:46 And I still haven't put my limits in, just tidying it up as 1 of 3.

07:51 When you put your limits in, put three in first then minus one from it -- not one, minus the function with one in it.

07:58 So we can rewrite this as 2 root 3.

08:01 And then I'm going to take away 2 root 1.

08:04 So I've just replaced those Xs.

08:07 When you put this into your calculator, you will see that you'll get answer of 1.464.

08:14 So, you can see how close they are really up to one decimal place there exactly the same 1.4 and 1.4.

08:21 They're a little bit different from the second decimal place on, but that makes me really believe in my trapezium rule formula that I've done it correctly.

08:29 Obviously, it's an approximation so it's going to be a little bit out, but you can always double check with your general faster method or you can check your faster method if you have time by using the trapezium rule and any amount of ordinates really.

08:44 The more ordinates you have, the more strips you have, the more accurate your answer will be because they'll be thinner and you'll waste less, less area.

08:53 It will be more accurate.