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(a) approximate the value of each of the given integrals by use of Simpson's rule, using the given value of $n,$ and $(b)$ check by direct integration.$$\int_{0}^{2} x \sqrt{x^{2}+1} d x, n=4$$

Calculus 1 / AB

Chapter 25

Integration

Section 6

Simpson's Rule

Integrals

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So hello everyone today. We're going to solve the integration from 0 to 2 X. Into the whole route of expert plus one where N is equal to four. In the good question. We are told to integrate this first for party using the Simpson's rule and find the integral. And for part B we need to use simple integration to check the answer for party. So let's start with party. The former for symptoms through is given at the top is is the integral from A to B F X G X approx mr Dell dx over three into fx not plus four. Fx one was to F X two plus +463 So it alternates between four and two and Fx And so except for the first and the last term, everything else is um multiply before and to an alternate being four and two. So to find little X. The formula is B minus A. R. N. And the government question B. S two which is the upper limit And is the door handle a mentor to zero Or end given n. s. four physical is still half. So now that we have the X. As half we can find the extremes. So X. Note will be A which is zero as it starts with zero annex one. As Uh huh. As it increases behalf every time extra would be one extra would be 312 X four. We can write their property export would be to do which is be no we have the um X points. We need to find the via points by plugging in the X. Points into the main equation which is X into root of X squared plus one. So when you plug in the first F. X. Not you get as um zero entered route of zero plus one that gives you zero. Um Then I'm gonna skip a step and directly find four X. One instead of just fx for fx one instead of just fx one. So let me just write that will take I'm gonna find four into affects one that it's given in the formula. You can see it's for it affects one as a second term. So four and two X. one. When you put half into the main integral equation you get do exercise so off. Uh as I wrote fire Then f. f. x. two is four. Okay that's right too because the third time that I went to into the earth of X. One and then the third time the next door. And that's the answer is stupid too. Then FFX three it's the third time as for into the death of three by two Which gives you the answer is 3-13. Sorry for the background as if you can hear that and finally affects for is just of two which you get the answer as to route five. So now you have all the values zero Route 5-2-3 13. Now you need to plug it in the equation. So the formula is Delta X over three. I'm gonna write that. So delta X over three. We have given the death tax as one or two. So one or two Into three gives you one or 6. one or 6 into zero plus book five. Keep in mind that we have already moved there for and to an alternate between four and 2 to find the our final terms which have to be plugged into the equation. So that's what we're doing right now. Last two or 2 Plus three or 13 plus to route five. Now that you get the answer as But you get the answer is 3.39-2. And that is your final answer for part A. And that's what you get after applying Simpson through. Um Now for party we need to prove this by prove that this answer is correct by finding the integration definite integral. So it's definitely went from 0-2. I'm just gonna write it down. It'll pick X. O. X. Where this one. Now to integrate this you would use U substitution where you would be Ruutel X square. So on the right that town except this one so you will be. Uh huh. So to expect this one and uh you would get the um now that now that you've taken us out of X. Purpose one your balance you can change your upper bound and lower bound and you'll get the lower bound as one an upper bound as your five So on and five of you square plus t you. And then you need to integrate this and you get the integration as you cube for three from 15? Yes sorry not passive. Let's see you want to come here because it's a definite integral and um when you plug this into the equation you get five route five you can use the calculator to find the answer for this which is five or 5 or three. Sorry this thing is not working with uh 55 43 -1 or three. Which when you find it in the decimal farm We get 3.39. So you get the same answer as what he had gotten using the Simpsons room but this is definitely more accurate than Simpson role. So that's how you check from normal integration that your Simpsons will did work. And so I hope you guys understood that the main part of this question was finding the um, in technologies in some side room, uh, and knowing the formula for it. And I hope you guys understood that, and thanks for watching.

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