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'6[-/4.54 answered Calculate Description all the Points] parts 1 + y2 doogle tnteg Make integral: DETAILS sure dA nof follow 'x)} y) | SCALCET8 the 0 <...

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'6[-/4.54 answered Calculate Description all the Points] parts 1 + y2 doogle tnteg Make integral: DETAILS sure dA nof follow 'x)} y) | SCALCET8 the 0 <X<5, Instructions 15.1.515.XP set by 1 0/1 WebAssign Submissions when Used submittinSubmit AnswerView Previous Question

'6 [-/4.54 answered Calculate Description all the Points] parts 1 + y2 doogle tnteg Make integral: DETAILS sure dA nof follow 'x)} y) | SCALCET8 the 0 <X<5, Instructions 15.1.515.XP set by 1 0/1 WebAssign Submissions when Used submittin Submit Answer View Previous Question



Answers

$7-16=$ Use (a) the Trapezoidal Rule, (b) the Midpoint Rule,
and (c) Simpson's Rule to approximate the given integral with
the specified value of $n .$ (Round your answers to six decimal
places.)
$$\int_{0}^{3} \frac{1}{1+y^{5}} d y, \quad n=6$$

Were given a vector field F and a curve that is oriented clockwise is viewed from above sea and were asked to Stokes his serum to evaluate the line. Integral oversee of F so f is thieve vector field one I plus X plus y zj plus X y minus root C, K and C is the boundary of the part of the plane. Three X plus two y plus C equals one in the first often. So first, let's calculate the curl of F. This is equal to X minus. Why I and then we have minus why j plus okay. And we have their surface s. You were told. This is bounded by the curve which bounds the portion of the plane. Three X plus two y plus Z equals one over. We're in the first often so the domaine de, which is the set of pairs X y such that being the first often X has to lie between zero and one third. And then why lies between zero and the line one half of one minus three x, and we want to orient s upward, since our curve is counterclockwise when viewed from above, and we have their surface can also be described as a function of X and Y. So this will simplify her calculations a little bit. So Z equals G of X y, which equals one minus three X minus two y and therefore have that the line integral oversee of F This is by Stokes is there equal to the surface integral over s of the curl of F and evaluating This is the double integral over the domaine de of and here we're using the shortcut that I mentioned before we have the opposite of the X component of Sorry, the first component of our vector field f So the opposite of X minus y times the partial derivative of Z prostrated of G, if you like with respect to X which is negative three minus the second component of F, which is X plus y z times part of derivative of G. With respect to why which is negative two plus the third component of F, which is X Y minus root C d. A. And plugging in Z is in terms of x and y. This reduces to the integral from writing is an iterated integral from Mexico. 01 3rd integral from 0 to 1 half of three x minus one of simplifies to one plus three X minus five y All right, de y dx actually in the steak here. Sorry. This should be the components of the curl of F instead instead of the components of F. So this is instead of X plus y Z. This is negative y times negative too. And instead of x y minus root, see, this is simply one makes a little more sense. So we obtained this and then taking the anti derivative with respect to why we get integral from 01 3rd of why plus three x y minus five hats y squared from what I equals zero toe y equals one half of three X minus one The X and substitution gives the integral from zero toe one third. Um, and here we have one half times one plus three x Times three x minus one minus five halves times three x minus one squared times one half squared. So if I've had times of fourth times three x minus one squared DX and multiplying the south. We get the integral from zero to one third of Let's see, we have X squared term is negative. 81 8th x squared The ex term is 15 4th x, and the constant term is negative. 1/8 DX Taking the anti derivative We get negative 27 8th x Q plus 15 8th X squared minus 1/8 x from zero toe, one third Plugging in. We get negative 1/8 plus five 24th minus one 24th and this simplifies to one 24th.

Yeah. Cancer for Question 60 v oh equals integration from 0 to 1 floor. Rule off. Is it multiplied, boy e to the power minus zip diesel. Sorry, These it not the X and the end. The quilting? Yes. And using trapezoidal room and in equals pain a equals zero be equal One. When you get the value of Delta X equal one minus zero over 10 it was going toe. And so the trapezoid rule gives that t four equal 1/20 multiplied Boy, if you plus to if we're going to one plus do if who are going to lost to f 4.3 plus do if or going for plus two If 4.5 plus who is 4.6? Lost two. If a 4.7 lost two if or going eight lost two. If we're going tonight, Plus if one. And by direct substitution this value equal or 0.37 toe 99 for Section B and using the mid going through. Delta X also ate one 1 14 equal over in one the mid points off 10 sub intervals or 1/20 3/20 5/20 7/29 over 20 11/20 13 15 15 17/20 and finally 19/20. So using the midpoint through, we will get the value off every 10 equal one 14 if 1/20 plus Israel 3/20 plus if 5/20 etcetera. Yeah, still, if 19/20 and this value equal, we're going 380 pain 94 four Section C and using Simpson's rule working if it equal rolling one plus X squared and then equally just the X equals we're goingto in Simpson's rule, we obtain yes to them equal. 1/20 was the blind boy if zero loss. What if? For went one plus toe if for going to plus for it 4.3 lost through it or wait for plus for if 4.5 lost toe. If Hogan six plus four if 417 lost. Who, if open lost swore if it's for going tonight, plus if it him by direct substitution, this value required 4.3 76 three three zero

Can this integral? We're going integrate with respect. Thio X First there's no X in the inta grand, so we're really just integrating DX there. So we get five or 62 pi over two. The co sign of Y is a constant. So now we're going to integrate minus one of five g x de y, and you don't have to write it like that. But I want to do to see that you're just integrating DX. So get five or 6 to 5 or two. Go. Sign of why the integral of D X is X minus 1 to 5 D Y. So I get five or six toe five or two and a girl but co sign of why five minus negative one, Do you? Why it's like gives you six and then the integral of the coastline of why is the sign of why I Richard a five or six. So get six sign of pi over two minus sign of five or six. However two that's here. The Y value is one. So that's six times one those five or six years Juan my s five or six that's 30 degrees, 12 square to three. So that's one half one minus a half. That's a half halftime. Six. That's 30

Answer for question 68. Decoration from zero to buy four. Excuse I necks DX and in equal for the interval, wits will be Delta X equal by minus 4/4. Equal. Oh boy! 78 Life A. For the midpoint tool. We will calculate the values off the for the function at the middle off each interval. So we start at X equal. But boy over eight and increments Boy Idol, the X and IT tricks. We will get the value off FX at X equal. We're going to three nine if the X will equal or going 36 and X equal one going to 17 if x equal. 4.4 50 eight x equals 1.96 if x equal. Minus 4.75 and finally at x equal 2.74 if x equal. Minus 2.53 and before equal integration from zero by 46 Whose line X dx equal Dill takes multiplied boy, if x note. Plus, if it's one plus, they fix it glass. If accessory by direct substitution The value off informal equal negative 1.29 full of life. Seven. You equals X Do you equal d x d v equal design x dx ndvi equal Zion X So the value of integration from zero to buy Excuse I'm xdd X equal Ex Zain X minus Integration from zero to buy Zion x 80 x and this will equal X Zain X plus whose line X from zero to buy. The result will be boys I am boy Plus because I am bi minus zero Blasco Ziemba by direct substitution equal zero minus one minus one. So the final result will be minus two. Okay, if we can clear, the room will equal minus two minus minus 1.29 45 seven four will equal minus. We're going 05 42 for which and 60 for some integration and same interval with built X. Pick one by over four and using same control. We will calculate the values and the interval boundaries and h X We will get the value off FX at X equals zero effects equals zero at X equals 0.78 If X equals 4.55 at x equal 1.257 If exit 10 at X equal 2.35 if X equal minus 1.266 and finally at X Equal city one going to 14 Yeah, if x equal minus 3.14 Yeah, same business rule has in the plus one terms. Here we have five terms. So yes, before equal, there's the X by three multiplied boy if it snowed. Loss if, for a fix one lost do if exito loss for effective three plus it fix for and as a result, write lyrics. That institution will be minus 1.8 one boy three 1.985 six one one from birth Again, the value of the integration off extra cozy in x 80 x would equal minus two. So if we calculate the error so by subject, by subtracting the approximations from the actual value, there will be minus two minus equally minus or goingto 14


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