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Now, let us discuss a bit about basic integration methods.

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And here in this figure, I try to explain to you the problems that occur when we take two simple methods

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for integration.

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So as you probably know, when we integrate a function from boundary to boundary B, we basically try

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to calculate the area under the curve.

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So basically this red area here between the blue curve and the x axis.

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And if we try to calculate this integral starting from minus three to plus three, then when we do it

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numerically, we have to, first of all, find out some points of the graph, of course.

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So we must determine d the f of X values of these red dots here.

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And then what we can do is we can transform the integral, which is like a continuous treatment of this

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problem.

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We can transform it into a sum.

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And now the question is, how do we calculate the sum exactly?

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There exists several methods, and the most simple one would be to just consider basically a weighted

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sum.

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So basically, you take all of these values here and you add them up and then you just take into account

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the size or the distance between two of these dots.

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So basically the width of these bars and then you just add up all these rectangles here, and this will

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approximately give you the integral.

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However, you can see already a problem that occurs, especially when you have just a few of these points

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and also when the function doesn't go to zero at the boundaries.

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This is that the and the end point.

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So here, minus three and plus three are over pronounced because here we are not really calculating

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the integral from minus three to three.

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But we calculate the integral from, let's say, minus three point two five two plus three point two

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five.

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So this is a bit of a problem here.

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Our range is not perfectly determined.

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So better methods have to be found.

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And a very simple idea would be to just say that we can take the edge values of the interval over which

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we integrate.

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And this point here will only be taken into account with half the width or half the weight.

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You could also see.

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So you see this rectangle here has only half the width of the neighboring rectangle and then at least

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the interval the range over which we integrate is correct.

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And then there exist many other methods.

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For example, this truck has trapezoid methods, or we can also derive more advanced methods that give

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you a much better value for the integration.

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So let's get started and let's define the function and I have plotted here, and let's create the data

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points that we will use for our integration.

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And also, of course, let's first discuss the analytical results, which will be our reference point

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for evaluating the quality of these integration methods.

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So first of all, as always, we will load our modules are Nampai and Plotnick.

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And then we will define a function.

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So this will be the function that we want to integrate, and the function is 0.5 plus zero point one

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times x plus 0.2 times x squared.

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And then we have the last term 0.03 times X to the power of three.

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So it's a polynomial of third order.

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And if we want to plot it, we just write X list is in a line space and put in space.

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Do you see the plot range approximately, let's say, minus three point five to three point five and

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seventy one points?

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And then you plot the function by writing plot of plot x list and then the function acting on the X

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list.

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And here we go.

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This is our functions the same as is shown here.

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We can, of course, also change the Y range and we can add labels for the axes.

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I think it's OK to just leave it as it is because we have the plot already here.

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So now for the analytical solution, here we have again the function.

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And when we calculate the area from minus three to three under the curve, then we have to integrate

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over this function.

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And so basically we have to increase the power of the X and then take into account the factor here.

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So basically, when we arrived this function, we should end up with our initial function.

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So you can test this, the derivative is one five plus one over 10 x plus one over five x squared plus

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three over 100 x to the power of three.

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And then we have here the boundaries from minus three to plus three.

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So now we just have to calculate these two terms here.

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So I copy this line of code from my other notebook and then run it.

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So these are just these two terms.

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And the solution is six point six, and this will be the reference point for the integral of this red

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area.

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And then we want to consider here a situation where let's say we have our data could be some experimental

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measurement or could be just some.

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Some data points that we have from have taken from the internet of some function.

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Or maybe we did a survey or something like that.

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And so we don't know the function behind it yet.

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So we only know the data points that we are going to create next.

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So the data points aren't the ones that I have plotted here and read.

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This is the only information that we have about the graph.

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So we want to make the situation as difficult as possible.

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So the X points will be a in space and it goes from minus three to three and we have 13 points because

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then we have a distance of the neighboring points of zero point five.

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So you see here we have 13 points

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and then the data itself will be an every and detailed array and the X coordinates will, of course,

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be the X points and then the Y coordinates will be the function acting on the X points.

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So this will be our data and then we can just plot the data and this will just be PLG dot scatter data

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zero and data one.

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So this will be the situation that we will start with.

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We have just these data points and we try to calculate the integral over the corresponding function,

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which we don't know from the area minus three to three.

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This we will do by different methods and we will start in the next lecture.