solve-each-given-problem-by-using-the-trapezoidal-rule-a-force-f-that-a-distributed-electric-charg-2

Question

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 $k.$

Victoria Dollar · Accepted Answer

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&#x27;s rule. And then we&#x27;re supposed to go back and find the exact value by integrating it. So And we&#x27;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&#x27;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&#x27;s going to give me 2/4 or it&#x27;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&#x27;m going to get to If I plug in one half into X. I&#x27;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&#x27;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&#x27;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&#x27;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&#x27;m going to get 49. And so this is going to give me 12.25 Okay, so that&#x27;s the trapezoidal rule. And then we&#x27;re gonna also do this by Simpson&#x27;s rule. And so Simpson&#x27;s rule is kind of similar to trap his idol. So it&#x27;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&#x27;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&#x27;s not committing multiplied in front of it. So for mine, it&#x27;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&#x27;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&#x27;m going to keep this point 5/3 just because of the fact that it&#x27;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&#x27;m going to take my X squared, and I&#x27;m gonna add one to the exponents so it&#x27;s not gonna be a three, and then I&#x27;m going to take the three and divide by three because remember, actually, we do it this way, so I&#x27;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&#x27;re going to cancel some have X to the third plus two x from 2 to 0, and I&#x27;m going to do the two&#x27;s first. It&#x27;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.

Yuou Sun · Answer

problem. You have it. There&#x27;s no tricks here. Just calculation, right? You need a calculator. Here&#x27;s a function and their tactics equal Tau four minus zero, divided by age toys. Right There actually is one. You need to know what is P. M and asked. People don&#x27;t know. Just look at the book, right? Here&#x27;s a T on asked. There has is equal to 1/2 and everything is already known, so just calculate you could get a correct answer here.

Chris Trentman · Answer

were given an integral in the value of n and in each part where asked to use a certain rule toe. Approximate this interval with this value event. Okay, the integral is the integral from 0 to 4 of the natural log of one plus e to the X. I mean where n is equal to eight in part a were asked to use the trapezoidal rule. So first for this rule you want to determine the interval lengths Delta X These are be right most endpoint four minus the left most endpoint zero over n, which is eight. This is 4/8 or in a decimal notation 0.5. The next step is to find required function values. So in trapezoidal rule, we want to find the values of the function on the end points of these sub intervals. So I&#x27;ll make a table on the left side, we have the end points X and on the right side we have the value of our function, which is natural log of one plus e to the X mhm. Let&#x27;s start off are left endpoint zero and will increase by Delta X or 0.5 until we reach four. So we have zero 0.51 point 1.5 and so on. Mhm. Now, for the right side, all we have to do is simply plug in attention X from the left side. So, for example, plugging in X equals zero. You have the natural log of to for our first entry in decimal notation, this is to eight decimal places 0.693 14718 Likewise, we plug in X equal 2.5. We get approximately 0.974076 98 We continue in this fashion all the way to X equals three. And so your table of values should look something like this paying and how accurate want to be. You may have more or less decimal places on the right. Now that we have these values, we&#x27;ll use the formula for the trapezoidal rule for approximation, which will call T eight for trapezoidal. We know this has n plus one or a total of nine terms, and it&#x27;s some and that t eight is equal to our formula is the in the interval length felt x over two times f of the first point x zero So this is zero plus two times ffx, one plus two times ffx, too. And we continue in this fashion all the way up to two times ffx seven and then plus just one times ffx eight, which is three. And then we simply plug in the values for Delta X, which is 0.5, and all of our values from the table into the some in the answer is approximately 8.804227 to six decimal places. That&#x27;s the answer for Party now in Part B were asked to use the midpoint rule. This is similar to party, except for in the mid point rule. We have the same steps size Delta X equals five, but now we want to find the values of the function natural log of one plus e. D. X at the midpoint of these intervals. So we&#x27;re going to start drawing another table. First column X second column is the natural lot of one plus e to the X. Yeah, and instead of starting with X equals zero, the left endpoint we&#x27;re going to start at the midpoint of the first sub interval, 0.5, which is 0.25 Then we&#x27;re going to add five until we reached the midpoint of 3.540 which is 3.75 So we have 25 plus 250.5 point 75 plus five is. Mm hmm. 1.25 plus 0.5 is 1.75 And so on, then, just as in part A, we&#x27;re going to plug in the values and left to get the values on the right. So, for example, plugging an X equals 0.25 and two natural log of one plus e to the X. This is approximately 0.825939 42 However many decimal places you want, I would suggest a least six. Likewise, we plug in 0.75 We get 1.136 871 and I&#x27;ll continue this all the way down the table. Table of values should look something like this. And finally, we&#x27;re going to use the formula. So we know that the midpoint rule our approximation will call M eight. We know that this is going toe, have the total of n or eight terms in its some and the formula is the N eight is the length of a sub interval Delta X, which is 0.5 times the values of our function at the midpoint of each of these intervals. So we have from the table above f of x zero plus f of x one plus all the way up to F FX seven. So we have a total of eight terms in our son. And if you plug this into your calculator, this is approximately 8.799 212 to six decimal places under part C, whereas to use Simpson&#x27;s rule approximately interval integral I mean now Simpsons World actually pretty similar to the trapezoidal rules. I&#x27;m going to overwrite heart a here, so this now becomes part C just is in part a and part B. The interval length is 25 and also just is in part a. The table is going to consist of on the left all the end points of the sub intervals on the right, the value the functions at those end points. So this table is the same as in part egg. But now our formula is going to be a little bit different, so below party here, I&#x27;ll finish part C. So we know that s a r approximation giving substance rule as n plus one for nine terms just like part A and nine terms. But we know that these form of S eight is going to be the some interval with Delta X over three now instead of two times f of x zero plus four times value of after the next and point X one plus two times the value that for the next 10 point at thanks to plus four times that I have effort the next endpoint x three. We continue in this fashion, alternating between four and two until we get all the way up to four times half of x eight. Sorry, it should be four times f of x seven plus two times epic Lexi. Well, it&#x27;s not really four times after that. Seven since sevens on. And then just one time different X eight puts the last turn. It&#x27;s a total of nine terms in ourselves. And so if you plug this and do a calculator using the values from our first table up here, this is approximately 8.804 229 out to six decimal places. Okay,

Victoria Dollar · Answer

Okay, So for this problem we have in equals for and we have the function f of X equals, um two X squared plus one. And we&#x27;re on the interval between zero and to. So what we want to do is we want to figure out we want to create a table of all of our values. So x and f of X and we want to figure out how maney what the increments we&#x27;re going to be. So we&#x27;re going to take to minus zero, which is our interval and divide by four. So we&#x27;re going to get one half, so I&#x27;m going to do zero 0.51 1.5 and two. So if I were to plug in zero into f of X, I would end up with one 0.5 as X would give me f of X equals 1.5 when x is one f of X is three one x is 1.5. Ffx is 5.5 And what exes to ffx is nine. So now that I have that, I can go ahead and do the trapezoidal role. So the trapezoidal rule for this function is going to be a t of four equals. Um and then we&#x27;re going to do one half over to And then what we&#x27;re going to do is we&#x27;re going to do our f of X is on each of them. The outside, too. We&#x27;ll just be that f of x value. So this one will be one, and then all the inside ones will be multiplied by two. And so I&#x27;m gonna do two times 1.5 plus 2.3 times three plus two times 5.5 plus nine. Remember that outside one does not get anything multiplied by it. So then, if I so point 5/2 is going to be 0.0.25 than if I were to add and multiply all of these together, I&#x27;m going to get 30. So that&#x27;s going to end up being 7.5 and then for part B. So this is going to be, um, Simpson&#x27;s rule. So we&#x27;re gonna have s of four equals, and it&#x27;s gonna be point 5/3. And this one is similar to trapezoidal rule in the sense that it&#x27;s going to be the outside to We&#x27;re just gonna be ffx, and then the inside values. They&#x27;re gonna alternate being multiplied by four and two. So let&#x27;s start off with four and multiply it by 1.5 plus two times three and then go back to four times 5.5 and then the outside won again will not have anything multiplied by it. So I&#x27;m gonna have point 5/3. If I were to add and multiply everything together, I&#x27;m going to get 44. And so then this is going to be 7.333 repeating. And then last but not least, I want to take the integral from 2 to 0 of two x squared plus one DX. So I&#x27;m going to end up adding one to the exponents. And then don&#x27;t forget, I&#x27;m gonna multiply by the reciprocal of my new exponents. So then I&#x27;m gonna have plus one x from 2 to 0. So now I&#x27;m gonna have two thirds X to the third plus x from 2 to 0. So I&#x27;m going to go ahead and do my squared or do my two first. So, to to the third is eight. Um, I have 16 3rd plus two, and then I&#x27;m gonna have this will all be zero because I have anything time you have X multiplied by anything, it&#x27;s going to be zero. So then 16 3rd plus two is going to end up giving me. I believe it&#x27;s 22 3rd, which is the same thing as 7.3333

Yuou Sun · Answer

problems. 13 You need a calculator. Just speculation. There&#x27;s no tricks here. Their taxis equal to form on a sterile guarded by eight. So it&#x27;s one hunt, and you need to know what is he? A man asks. Here&#x27;s a T M s and then asked, Everything is already know in just multiply arisen tactfully. You got the correct answer here.

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Solve each given problem by using the trapezoidal…

Problem 20 Easy Difficulty

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Precalculus Review - Intro

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Catherine R.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus