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Approximate the values of the integrals defined by the given sets of points.$$\int_{2}^{14} y d x$$$$\begin{array}{c|ccccccc}x & 2 & 4 & 6 & 8 & 10 & 12 & 14 \\\hline y & 0.67 & 2.34 & 4.56 & 3.67 & 3.56 & 4.78 & 6.87\end{array}$$

Calculus 1 / AB

Chapter 25

Integration

Section 5

Numerical Integration: The Trapezoidal Rule

Integrals

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Let's approximate the value of the interval from 2 to 14 of why differential effects by the traps of the rule. And the set of points given here, we have 1234567 points in the plane for each value of X. We had the correspondent value of why that is wise function of X. That's why we see here why differential of X. We are not even a formula but a set of points. So we hope to to use the traps of the rule using this set of points. And for that the first thing we get to see or notice is that the values of eggs are separated, the consecutive values separated by a common distance. That will be a step size age. And that's the case because for example, from 2 to 4 we have two steps to difference of two but it's the same difference from 24 to 6 And from 6 to 4 from 8 to 10 from 10 to 12 and from 12 to 40. So we have always A step of four between two consecutive notes in the table for the X values. And that is the value of age. So the first thing we notice is ages too. Now we have the europe correspondent images for all the notes. The other thing we notice that the first note is just the left hand point of the interval of integration or the lower limit of the integral. And the last note is indeed the last the upper limit of integration or the writing point of the interval of integration. So these are the things that get to feed in order to apply through beside the rules. So we have this and now we know that the integral From 2 to 14 of white differential x will be approximated by T. And the number of serve intervals are 1, 2, 3, 4, 5 and six. Because we have here again one interval here, Another one here, 2, 3, 45 and six of intervals. So in equal six. So to six in this case it's given us a judge half times. And in this case we're going to use the notation of why that is why zero which corresponds to the first image or the image was the first note. This one here Then plus two times The sum of why Soup I from I equals one two In this case 0123456. So 25. These are the internal notes. Hear why someone Why too? Why three x 4. NY five and plus the last images. Why six. So this is the formula of the traps of the rule with six of intervals. Using the notation of wise of art. That is The image is given in the table starting at Index zero for the first one here and Ending Index six. Last image here. So putting the values you will be Age. There is 2/2 times 0.67 which is what zero here plus two times Then we have 11 is this one here to point 34 Plus Y two is 4 56 Plus or three is 3.767 Plus. What four is 3.56. Ny five is 4.78. These are the internal the images of the internal note And that's why they are some together and multiply by two Plus the image of the last. Nowadays 6.87. So here we have one. So it ended up being 0.67. Glass 6.87. It was seven 54 plus two times. This sum here is equal to 18 yeah 0.91. So this is seven 54 plus 1881 comes to 37 point 82. And finally there's some here is equal to 45 point 36. So The integral from 2 to 14, You know from two 14 of y differential of x Is approximately equal to 45 point 36 calculated yes. Inter upside the rule and the set of points in the table. It was fundamental in order to apply the so the rules that the first note on the table, the first value of the law was value of ecstasy. Um The whole limited integration. The greatest value of X in the table is the upper limit of integration. And then there is a common distance between any two consecutive notes or values of X when they are ordered from uh in ascending all order us in this table, that is the distance between any two consecutive nerves. Got to be the same. It's not we cannot apply the episode the rule in the classical way there it is defined. Maybe we got to do some other things to apply in terms of the rule where the notes are not equal space but to apply that we need this condition to be fulfilled in the table of data. In this case is this true? That is we have the end. The lower and upper limits of integration are the first and last values of X when it is order in ascending order. And the distance between the two consecutive notes is to in this case it's the same value all the time, that's the value of age, the step size And then we count the numbers of intervals and equal six in this case. And we can apply the formula of the top side the role with equally spaced sub intervals.

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