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Solve each given problem by using the trapezoidal rule.A force $F$ that a distributed electric charge has on a point charge is $F=k \int_{0}^{2} \frac{d x}{\left(4+x^{2}\right)^{3 / 2}},$ where $x$ is the distance along the distributed charge and $k$ is a constant. With $n=8$, evaluate $F$ in terms of $k.$

Calculus 1 / AB

Chapter 25

Integration

Section 5

Numerical Integration: The Trapezoidal Rule

Integrals

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Okay, So for this problem were given that in equals four and that f of x equals three x squared plus two and were asked to find out the area under the curve estimation wise, um, using the trapezoidal role in Simpson's rule. And then we're supposed to go back and find the exact value by integrating it. So And we're also given that this is between that X is between two and zero. So what I want to do first is I want to break in down, So I want to go between, um so I kind of want to break this down. So I'm gonna do X and f of X, and I want to break this down between zero and two. So I want four steps. So if I look at this and due to minus 0/4, that's going to give me 2/4 or it's gonna be one half, So I want to do this in in intervals of one half. And so if I were to go back and plug this in s 05 X equals zero and plug this into ffx, I'm going to get to If I plug in one half into X. I'm going to get 2.75 for ffx. Um, f of one is going to be five f of 1.5 is going to be 8.75 and then f of two is 14. Okay, so now what I wanna do is I want to do the trapezoidal rule. So what I need to know is the formula. So it's going to be teeth in equals the change in X over to of F of X of zero plus two f of X one plus two f of X two and then so on and so forth. And then plus two f of um x of in minus one. So we have to all in the middle, and then the two end values are going to be just without a two multiplied in front of them. So since in equals four for this one, we're gonna have t four equals. Our change in X, which is 0.5 over to my f of X zero we said was, um was too. F of X one is 2.75 f of X two is going to be five f of X three is going to be 8.75 And then, of course, my last one is 14 which I'm not going to multiply by two. So then, after I So I know that point 5/2 is 0.0.25 And then if I were to multiply and addle of these together, I'm going to get 49. And so this is going to give me 12.25 Okay, so that's the trapezoidal rule. And then we're gonna also do this by Simpson's rule. And so Simpson's rule is kind of similar to trap his idol. So it's gonna be, uh, changing X over three and then f of x of zero plus four f of x one plus two f of x two plus four f of x one So on and so forth. And then, of course, my last values it's gonna alternate between four and two in between the first and last and then the first and last is just gonna be time without or multiplied by one. So there's not committing multiplied in front of it. So for mine, it's gonna be s four equals point 5/3, so f of X zero is, too, that I have f of X one, which is 2.75 And then f of X two is going to be, um, five. And then I'm gonna alternate with the four against, and then my next one is going to be 8.75 and then my last one is just gonna be 14 without anything multiplied to it. So then I'm going to keep this point 5/3 just because of the fact that it's not gonna be a rounded decimal. If I add all of this together, I get 72. And then my final answer I get is going to be 12. Exactly. So now we want to integrate. So when I do the integral from 2 to 0 of three x squared plus two d x. Okay, so if you recall, I'm going to take my X squared, and I'm gonna add one to the exponents so it's not gonna be a three, and then I'm going to take the three and divide by three because remember, actually, we do it this way, so I'm gonna multiply this by one third. Remember, I wanna multiply by the reciprocal of the exponents. And then, of course, to is simply gonna become two X. This is gonna be between zero and two. So if you notice my three and one thirds, we're going to cancel some have X to the third plus two x from 2 to 0, and I'm going to do the two's first. It's gonna be two to the third, plus two times two minus the zeros. So zero to the third, plus two times zero. So if we notice to to, the third is going to be eight plus four and then minus zero, so that just cancels. So this is going to be 12.

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