Integration Method: Example 2

00:01 Moving on to our second example.

00:03 You may find this a little bit familiar because we've done this previously when we used the trapezium rule.

00:09 So we applied the trapezium rule on this question with five ordinates and we found an area.

00:15 Hopefully, we can now do this faster using integration and we should get a more accurate answer to what we got earlier with our approximation.

00:24 Let's have a look at our integral.

00:26 We have 0 to 1 and we have 3x plus 1 to the power of 2 dx.

00:35 Right.

00:36 This is a little bit more complicated than the previous question, because you think about it, when we were talking about differentiation, you have a function inside of a function.

00:46 But we now have to integrate this.

00:48 So, if you ever need to integrate something like this, you're still just trying to do the opposite of the chain rule.

00:56 Let's talk about this for a moment.

00:59 Let's make some space here.

01:01 When you integrate this, you integrate the outside function first just as you did with differentiation.

01:08 So when you integrate the outside function, you will get 3x plus 1 as it is without changing it, add 1 to the power and then divide by the new power.

01:18 So that's pretty straightforward.

01:20 But then remember, when we applied the chain rule, we multiplied it with the differential of the inside.

01:26 So with the chain rule, we multiply it with the differential of the inside.

01:31 What's the opposite of multiplication? It's division.

01:34 So in this case, we're going to divide with the differential of the inside.

01:38 So the differential of the inside function is 3, and instead we're going to divide it.

01:43 So you can see that one of the 3's comes from the power, and the other 3 comes from the differential of the inside.

01:51 I put those square brackets there just so I could remind myself of the limits.

01:55 But before I move forward to the limits, let me just put what I've just said as a formula on the side.

02:01 If you have to integrate f of x to the power of n, so I'm trying to create a function of a function here.

02:09 So we've got a function and a function inside.

02:13 You do exactly the same thing.

02:14 You add one to the power.

02:16 So leave the inside as it is.

02:18 Add one to the power.

02:20 You divide by the new power.

02:23 But then you also divide by the differential of the inside function.

02:27 So the differential of the inside function, let's just call that f' of X.

02:32 So that means just whatever is inside, we differentiate it.

02:35 And because I've set this as an indefinite integral without any limits, I just put plus C at the end.

02:41 So, if ever in doubt and struggling with a chain rule type, a function of a function looking expression with integration, just come back to this rule and hopefully this should help you out.

02:55 Coming back to our question.

02:58 I'm going to tidy this up before I do anything else.

03:00 I've got 3x plus 1 cubed over 9 between the limits of 1 and 0.

03:08 We haven't really put any limits in yet, so let's do that for the first time now.

03:13 When you put limits in, you put the bigger limit in first and then minus the smaller limit.

03:19 So, what I'm going to do is I'm going to put one in first.

03:22 So I'll do 3.

03:23 Instead of the X, I'll put 1 from here.

03:26 So, 1 plus 1 cubed over 9.

03:32 And then if I take away my second bracket, in my second one I'll put zero in.

03:38 So I'll do 3, 0 plus 1 cubed.

03:42 So everywhere that you see an X, you change it to the limits.

03:46 So in the first bit of my question, I just put one in.

03:49 In my second bit of my function, I put zero in.

03:52 So you do it twice basically but with two different limits.

03:56 Let's work this out so that it gives me 4 cubed over 9.

04:02 When I add this together, so 3 times 3 plus 1 gives you 4 cubed.

04:07 And then here, my second bracket, gives me 3 times 0 which is just 0 over 1 over 9.

04:14 If we work this out, this gives me 64 over 9 minus 1 over 9.

04:19 You can take 9 as your common denominator, and then you've got 64 minus 1 to give you 63 over nine, which you know is easily simplified so that gives us 7.

04:31 And remember that that is now our area.

04:33 So you could write this as 7 units squared, whatever.

04:36 I would say if it was centimeters or meters, you can just use this meter squared.

04:41 But the other thing to remember is that we use exactly the same question when we did the trapezium rule.

04:46 So, if I remind you what answer we get when we did the trapezium rule, using the trapezium rule, we got an area of approximately 7.0937.

04:58 And you can see that it's very close.

05:00 We've obviously lost a little bit of accuracy, and we only used five ordinates.

05:07 But the more ordinates you use in that method, the more accurate your answer would become.

05:12 The idea of integration is that you make those strips or those trapeziums or any shape there using so thin that they're almost infinitesimally thin, and so you add an infinite amount of them to get your area.

05:27 And this is what this integration has done.

05:30 It's a lot more accurate and it's a lot more precise and it's really quite incredible how it works, but the area that we've got is 7.

05:38 The area that we got using the trapezium rule was 7.0937.

05:43 It's very close.

05:44 You could make it more accurate by using more strips.

05:47 But, what I'm trying to say here is start to trust our easier integration method.

05:53 It's faster than using the trapezium rule and to believe that it actually gives you an answer which is very close to an approximation.

06:00 And so to summarize, we have just looked at the trapezium rule method and also the faster method.

06:07 I'm just going to go over the faster method when you're dealing with any type of function that you need to integrate.

06:14 So integration states that if you have something x to the power of n, you add 1 to the power and you divide by a new power.

06:22 If it's a function of a function, you add one to the power divided by the new power and then you also divide by the differential of the inside.

06:31 How about you try some questions now? So using the rules and the methods that we've spoken about, try these three questions.

06:39 The first one is just standard integration starting you off.

06:43 The second is quite similar to the function of a function integration problem that we dealt with.

06:49 And the last one does use the trapezium rule, so you will need a calculator.

06:54 But follow the rules step by step and hopefully we'll get to the same answer.