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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.$$\int_{0}^{2}\left(1+x^{3}\right) d x, n=2$$

Calculus 1 / AB

Chapter 25

Integration

Section 6

Simpson's Rule

Integrals

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So has everyone they were going to solve migration from has to X cubed plus one dx. Where N. Is able to do using the same syndrome, which is for part A and part B. We need to use normal integration and see which one is more accurate than what answers to get. That's the question. So there's symptoms through formula is going right here uh threatened it down. So integration from A to B. F X G X X cubed over three affects 04 plus four X one plus two X two plus four fx three. So it alternates between four and two except for the first time in the last term is affects zero and affects them. So to find the X went to b minus A or an and this question B is given as two minus zero, A zero or N. Which is to. So our delta X. Is one. So dietetics is one and because of that X is one are um we have that. Yeah 80 So it goes from a the first time will be the terms would be two than one than to. So those are the three times we need to find. So it goes on 0 to 2 so 01 and just but you don't need to find and so affects need to find Fx zero which is yeah of Syria which is putting it into the farmer. We get one. So when you get here in this we get one and Fx one difficult. A second term should be for into effects one which is one and you get the answer as eight. And then last time that Fx two now you won't multiply it too because the last term as I told you the last times would not be much better by four or two. It should be uh Insurance avenue multiple. It shouldn't be much better than anything. Not much and so it will just stay as effects too. And he at the third is to sew that up too and actually give you nine. No we just need to get in the phone love until make a line here so that the X. Over three so 1/3 into one last eight last nine. And this you'll get as six so around six so it runs after six. Yeah. Um And that's your final answer bitch. Well fired. Okay as around six. Now we need to use normal integration to prove this the party I want to do it in a different color of into integrate using normal integration so the integration of you know to do execute plus one. Oops industry that quick that's fine. Now integration of execute this one as X. Three plus one over three plus one plus X. From here to to that gives you X rays to 4/4. That's X. Now when you plug in to are you just aren't that real quick to raise the four or four us too and you subtracted by zero. You get the answer six and that was your final answer for part beef. So you get caught on. The answer is six ft both. This is rounded up to six so the finance will stay at six. So I hope you guys understood the Simpson drove. This is the main formula and we need to find out the extra and then put it in terms of effects um, and apply the four and two as multiplying it with it. And that's how you get your finances. Have you guys understood that? And thanks for watching.

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