Integration Method: Example 2

by Batool Akmal

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    00:01 So after that exhaustive method of using the trapezium rule, let's see if we can find a faster way of doing this.

    00:08 I'm going to teach you the standard method of integration, which is basically just integrating but backwards using our differentiation methods but going backwards to integrate.

    00:18 We're going to talk about all the notations here as to what it means to integrate and how you integrate and then how you can find values of areas under curves.

    00:28 So, let's start with just looking at an example curve.

    00:34 So say, for example, I have a curve.

    00:40 And let's call this f(x).

    00:45 If I wanted to find the area under this curve, I would express that as an integral.

    00:50 So to us mathematicians, this sign here is the integral which basically means integrate, find the area.

    00:57 That little integration sign usually goes next to the function, so any function that you were trying to integrate.

    01:04 And if it is a function of X, we can put dx at the end which means with respect to X.

    01:10 So we're integrating a curve with respect to X.

    01:14 Now, do you remember when we did differentiation, we found a general gradient.

    01:18 So we came up with the general gradient.

    01:20 And to find specific gradients, we then substituted numbers in.

    01:25 So when you do something like this, this would give you a general function for the area.

    01:30 So let's just call this F(x) for now.

    01:32 Let's call it big F(x).

    01:33 And this is a general function for the area.

    01:44 If you want to know a more accurate or a definite answer, you will write this as f(x) dx, but this time you would put limits on the side, so similar to what we've seen with the trapezium rule.

    01:58 And that basically means, if I was to do this function here, that means find the area everywhere.

    02:04 So we're looking for all the area underneath this curve, however, or just the function for all of the area.

    02:10 When I put limits in B and A, so for example if we say that B is here and A is here, if I do that, I am then drawing straight lines up to my curve and I'm looking for a very specific area here.

    02:29 So, remember that these are two different types of integrals.

    02:32 They both do pretty much the same calculations but they are different in what they tell us.

    02:38 This here is known as an indefinite integral.

    02:44 And this here is known as a definite integral.

    02:49 A definite integral usually has limits on it and an indefinite integral is just a general integral function.

    02:56 So if I was of do my definite integral and say that I've found my expression for the area, F(x), we can then put brackets around it to say that we need to substitute values A and B into it.

    03:10 When you put B into this function, it will find all the area here before B.

    03:18 But if you want to take away the area underneath, if you want this and if you want to subtract this area, you need to find the area from zero to A as well.

    03:28 So you can then minus F of A to get your answer as a definite value or to get just this black shaded area here.

    03:43 So those are just the basics and the notation.

    03:45 The sign for integral is this little curvy sign here.

    03:49 Any function then goes in the middle.

    03:52 The dx here just means with respect to X, you could integrate with respect to Y or other functions as we did in differentiation.

    03:59 We're going to end up with a general expression for our area.

    04:05 And to make that expression more specific, we're going to have to substitute values in A and B.

    04:11 So remember when we did differentiation, we found a general gradient.

    04:14 To find more particular gradients, we had to substitute Xs in.

    04:17 This is pretty much similar sort of methods here.

    04:22 Now, the next question you might be asking is, "How do we actually integrate?" So we know what it's doing, but what mathematics do we use to actually integrate? In order to do that, I'm just going to write that here.

    04:33 So imagine we're integrating a very simple, a very simple function.

    04:39 So say we're integrating x to the n dx.

    04:45 I'm not putting any limits here yet.

    04:46 We'll just treat it as an indefinite integral.

    04:49 Now, I'm just going to teach you this by thinking of differentiation.

    04:52 Because I said it's the opposite of differentiation, we're just going to learn the method by looking at the differentiation method.

    04:59 So remember what we said when we differentiate it, x to the n and dy by dx.

    05:04 We brought the power down and we decreased the power by 1.

    05:08 Integration is trying to cancel that out.

    05:10 So everything that we did there, we're trying to move backward.

    05:13 So we're trying to do counteract the -- bringing the n down and decreasing it by 1.

    05:20 So, in order to integrate, we're going to start with the power, where you're going to say x to the power of n plus 1.

    05:27 So you add one to the power.

    05:29 And if you think about what's going to cancel this little bit here, you divide by n plus 1.

    05:37 So, the two little steps to learn here when you're integrating, add one to the power, divide by new power.

    05:43 Remember, when you differentiated, you brought the power down, you minused one from the power.

    05:48 This time you're adding 1 to the power and you are dividing by the new power.

    05:54 The only small thing you need to make sure of, that when you integrate, you always put a plus C or a constant at the end.

    06:02 And remember, so many times we've done differentiation where you may have a number.

    06:07 So let's say you may have a plus C here at the end, and when you differentiated it, that disappeared.

    06:14 Remember that.

    06:15 So when you differentiated plus 7 or plus 10, it just disappeared to 0.

    06:20 We're now making up for that.

    06:22 So because it cancelled, we're now adding into it, so rather than any number, rather than a function or a constant at the end going to zero, we are now adding a constant to it.

    06:34 And this is how you integrate an indefinite integral.

    06:39 If, for example, I now wanted to put in my limit, so say we have b and a, x of n dx, you do exactly the same thing, you add 1 to the power divide by a new power, you then have the plus C at the end.

    06:54 And then you are now going to put the numbers a and b into this.

    06:59 So B goes in first because it's bigger, like we said here.

    07:02 And then A goes in next because it's smaller and then you subtract it to get your answer as a number.

    07:10 There's a lot of integration here on this one slide, so pretty much all of the basic integration methods that we've just put in one place.

    07:17 But, to consolidate these ideas, we now need to start practicing different types of questions.

    07:23 So let's have a look at some numerical examples.

    About the Lecture

    The lecture Integration Method: Example 2 by Batool Akmal is from the course Basic Integration.

    Author of lecture Integration Method: Example 2

     Batool Akmal

    Batool Akmal

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