5

Evaluate the line integrals $$int_{C} f(x, y) d s, quad int_{C} f(x, y) d x, quad ext { and } quad int_{C} f(x, y) d y$$ along the indicated parametric curve.$f(x, ...

Question

Evaluate the line integrals $$int_{C} f(x, y) d s, quad int_{C} f(x, y) d x, quad ext { and } quad int_{C} f(x, y) d y$$ along the indicated parametric curve.$f(x, y)=x+y ; quad x=e^{t}+1, y=e^{prime}-1,0 leqq t leqq ln 2$

Evaluate the line integrals $$int_{C} f(x, y) d s, quad int_{C} f(x, y) d x, quad ext { and } quad int_{C} f(x, y) d y$$ along the indicated parametric curve. $f(x, y)=x+y ; quad x=e^{t}+1, y=e^{prime}-1,0 leqq t leqq ln 2$



Answers

Evaluate the line integral, where $ C $ is the given curve.

$ \displaystyle \int_C (x/y) \, ds $,

$ C: x = t^3 $,
$ y = t^4 $,

$ 1 \leqslant t \leqslant 2 $

Okay. So what you want to do in this problem is you want to change everything to T. So you're gonna put tea in place of X. T. Squared for whitey cube for Z. And then dy since why is T squared Dy will be to T. T. T. All right so you get T. T. Squared. Eat the T. Square times T. Cubed times to T. D. T. And he's going from 0 to 1. Okay so you have to integral de Erdogan E. T. To the fifth Times T. T. Square T. T. to the 4th DT. Yeah some substitute here. I'm gonna let you be T to the fifth then d'you is five T. To the fourth D. T. And if T. is zero U. is zero to the 5th which is zero If T. is one You is 1 to the 5th which is one. Okay so this turns into two. Now I'm doing Uz Erdogan eat to the U. Oh but forgot I need five T. To the fourth. And I just have T to the fourth. So I'm gonna put five in there and 1/5 out the front. So I have two fists. E. To the U. D. U. The integral will be to the U. E. To the U. From 0 to 1. So 2/5 it's the one minus E. To the zero Or to 5th E -1

Evaluate this line to grow, so we just have tickles. Won't want to. Four wise t X is one over two square root of TD thes t square The wise DT Xs square of t disease to tt so one, two, four This is one over to t to the one over two plus t Square Prostitute to the This should be three over to TT and we find the entire derivative t to the Stree Over to community property to over three. Four t to the Cube over three. Tito the five over to beautify by two over five in front. So for over five, then we probably before we had for the Israel with Who's eight for Cuba's sixty four four to the five over two is to to the fifth Power, which is thirty two thirty two times for his uh, one twenty eight minus one over three plus one over three plus four or five and you're left with fractions. Simplification, which I think I've had. Leave it to you. So I'll have to open another page

So if this question were passed to determine the lining to grow Uh, X Y yes. And we're given that the curve. We have a pyramid to repair amateur eyes curve where X is equal squared and wise. He would take two to t and we're told that goes from zero 21 All right, so the first thing we're gonna do is we're going to determine or clank for DS. Then we know that the S is square Heard of d y by d Squared plus DX by DT square TT So if we take why and we finest derivative with respect it about activity, so lies just to t so that the derivative of, uh, the derivative of that with respect to, uh, t so to t the derivative of to tea with respect a tease just to And then also we take a riveting vax with respect. Cities of excess just t square. The derivative of that with respected tea is simply too All right. So finally you get square root of two squared plus two. So this just four plus 40 squared. We can pull out of forest common factor the square root of forest to And then what's left inside this square and square root is just X squared. So we determined our ts, our ideas we have we know that access t squared. We know that. Why is two teeth so simply what we're gonna do is we're gonna plunk now everything back into this and try to determine. Okay, so now when we plug everything back, we get this big mess right here, but we're gonna try that still, So we pull out the tour. So two times two is four so we can pull that out as a constant And what we're left with tears. That's right. So we haven't too sweaty square, plus one under the square root, and we have a cubed on the outside. Well, let's try to solve this integral using substitution. So we're gonna sit you equal to the square, plus work, and then if we take the derivative of there we get to you is to teach. And then if we divide by two on both sides, we get that TV team is equal to you, divided by two. Also, we're gonna do something that might be a bit unusual, but, you know, we can also since we said u equals C squared plus one. What we can do is we noticed that he squared. Is he good to you? Minus one. So no, this is create because that we can write everything in the integral here in terms of you. So also, we were gonna change the limits so the lower limit was zero, but that we know that there's a mystical tissue we can pluck that into t squared +10 square plus one is just one. So that's our new limit. Because we want everything in terms of you and again, we have our upper limit off. The integral is teasing cool toe one. So if we plug that into use equal to the square Plus when we get one square, plus one more to So now we completely then we have everything in terms of you. So everything we have in terms of you and R d t is just Do you divided by two so we could pull that tutor the outside So four divided by two is just all right. So now we have to take the integral, uh U to the power of 3/2 minus you did well. The integral beauty, the power of three divided by two, is just due to the power 5/2 times Tool and the integral of U to the power half is disputed. The power 3/2 times two of us. And the limit is from 2 to 1 of these. So now we just plug everything back in. We don't forget we have a two on the outside. But then if we work out the out abruptly put this into our calculator, we're gonna get that. Our final answer is 1.2876 So that's the value of the line.

All right, So this question were acid Determine the mining to grow off X squared plus y squared. Plus they squared along the curb Access, equality Why is equal to co sign Titi and see is he was assigned to duty and we're giving the tea ranges from 0 to 2 pi. So the first thing we're gonna try to do is we're gonna try toe, determine the answer. This would lose share with the integral. Sorry, we're gonna write everything in the integral in terms of tea that were able to take me. So our only issue. So So extend your industy. Why can be written in terms of ts clubs on duty and he could be written in terms of ts scientist t Our only issue it with biz with DS or the click. But what we know the formula for the park length is DS is equal to the square root of d y by D T square was easy by DT Square plus plus the dx dy DT square plus D c biking square. All of them multiplied by. All right, So do you exploit eating? Use just the derivative off T but respected team which is just one square and d y by D. T is just the dirigible coastline to t I would respect a T which is just negative to sign t again. We're going to use u substitution. Use your substitution This and then the derivative off Easy by DT is just a derivative of scientific tea With respect again, you use your substitution. You get too close I to all right now you're going to square. These one squared is just one negative to sign to t swear it is Four signs square to a team and to co sign to t squared is four co sign square to And then now what we can pull out is we can float a floor is a common factor and we're left with science square to t plus coastline scored two teeth and then we know that this is simply one. So we have the square root off one plus full which is just route by Bt. So now finally we determined or ds in terms of teas. This Route five DT. All right, so now we plug everything back into our integral and then we get everything. So now So now what we notice is that we have co sign square to t plus sine square to do well, this is simply one. So we're left with t squared plus one and we can take the square root of five because it's a constant. And then pull it on the outside. So finally are integral. Look, something like this. So, Brooke, five times integral from zero to buy 50 square plus one detail. All right, now, the integral of two squared is too cute, divided by three and the integral of honest tea. And then you're gonna take the definite So this is a definite integral. So you're going from 0 to 2 pi. So we're gonna plug in to pie, and then we're gonna plug in zero, and then we're gonna take the difference of Well, the zeros are just just give us a zero and we have a square root of five on the outside. Two pi killed this eight by cubed, divided by three plus two pi. So finally, our line integral is very little five kind a pie que divided by three plus two


Similar Solved Questions

5 answers
Question Ii Predict_theprodualsLof the following reactions; choose trom A- the term that best describes the overall mechanism of the reaction. Electrophilic addition B. Electrophilic Aromatic Substitution Elimination reaction D. Nucleophilic addition Nucleophilic substitution (5 points)KCNCHONaOHCH;Brz/FeBtaNOzCHa NaOCHyMeOH H;c-C CHz-CH3Brz/HzO
Question Ii Predict_theprodualsLof the following reactions; choose trom A- the term that best describes the overall mechanism of the reaction. Electrophilic addition B. Electrophilic Aromatic Substitution Elimination reaction D. Nucleophilic addition Nucleophilic substitution (5 points) KCN CHO NaOH...
5 answers
0f 1If the current in the second col increases ata rale 0f 0.4C0 A/ Khat is tne magnitude of the incuced emfin tne first coil?Express your answer in millivoltsA2dRequestAnswerSubmitVz SWIU
0f 1 If the current in the second col increases ata rale 0f 0.4C0 A/ Khat is tne magnitude of the incuced emfin tne first coil? Express your answer in millivolts A2d RequestAnswer Submit Vz SWIU...
5 answers
Convert the expression x 3e2+2 iZel-iinto the form x Xete
Convert the expression x 3e2+2 iZel-iinto the form x Xete...
4 answers
If $A_{n}$ be the area bounded by the curve $y=(an x)^{n}$ and the lines $x=0, y=0$ and $x=frac{pi}{4}$, then for $n>2$,(A) $A_{n}+A_{n-2}=frac{1}{n-1}$(B) $A_{n}+A_{n+2}=frac{1}{n+1}$(C) $frac{1}{2 n+1}<A_{n}<frac{1}{2 n-2}$(D) $frac{1}{2 n-1}<A_{n}<frac{1}{2 n}$
If $A_{n}$ be the area bounded by the curve $y=( an x)^{n}$ and the lines $x=0, y=0$ and $x=frac{pi}{4}$, then for $n>2$, (A) $A_{n}+A_{n-2}=frac{1}{n-1}$ (B) $A_{n}+A_{n+2}=frac{1}{n+1}$ (C) $frac{1}{2 n+1}<A_{n}<frac{1}{2 n-2}$ (D) $frac{1}{2 n-1}<A_{n}<frac{1}{2 n}$...
5 answers
Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.$int frac{d x}{x^{2} sqrt{x^{2}+1}}$
Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.$int frac{d x}{x^{2} sqrt{x^{2}+1}}$...
4 answers
Predict which category the substituent (-S(CH )-/t below would be in terms of its effect on electrophilic aromatic substitution .electrophilic aromatic substitutionortholpara director; activator ortholpara director; deactivator meta direetor; activator meta direetor; deactivator
Predict which category the substituent (-S(CH )-/t below would be in terms of its effect on electrophilic aromatic substitution . electrophilic aromatic substitution ortholpara director; activator ortholpara director; deactivator meta direetor; activator meta direetor; deactivator...
1 answers
Practice: Calculate Vo in the circuit ofthe figure below using the superposition theorem . 16.630 sin St V02F1H2 cos [Ot AAnswer:4.63] sin(St 81.122) + 1.051 cos( [Ot 86.24") V:
Practice: Calculate Vo in the circuit ofthe figure below using the superposition theorem . 16.6 30 sin St V 02F 1H 2 cos [Ot A Answer: 4.63] sin(St 81.122) + 1.051 cos( [Ot 86.24") V:...
5 answers
Consider the vector space R? with the following non-standard addition multiplication: and scalaru +v = (4z,42) + (V1,Vz) = (u + Vi, Uz Vz ) ku = k(uz, U2) = (kuz, Uz) Which of the following vector space axioms does not hold?k(u +v) = ku +kv b) 1u = W c) u+v=v+u d) u+v€ R2
Consider the vector space R? with the following non-standard addition multiplication: and scalar u +v = (4z,42) + (V1,Vz) = (u + Vi, Uz Vz ) ku = k(uz, U2) = (kuz, Uz) Which of the following vector space axioms does not hold? k(u +v) = ku +kv b) 1u = W c) u+v=v+u d) u+v€ R2...
5 answers
NameDate#f(x) = 3x 2 and g(x) = x2 + 1,find f [g(-3)] and g[f (-3)]:What is the inverse of the following relation: {(~2,5), (0,4). (1,-8), (4,7)}What is the inverse function of the following: flx) = Sx-6What is the inverse function of the following: flx) = (2x + 1/28. State the domain and range of the following radical equation:
Name Date #f(x) = 3x 2 and g(x) = x2 + 1,find f [g(-3)] and g[f (-3)]: What is the inverse of the following relation: {(~2,5), (0,4). (1,-8), (4,7)} What is the inverse function of the following: flx) = Sx-6 What is the inverse function of the following: flx) = (2x + 1/2 8. State the domain and rang...
5 answers
Approximate f'(1) given that f(x) = 2x cos3x.-2.82670.2310-6.704255.6577
Approximate f'(1) given that f(x) = 2x cos3x. -2.8267 0.2310 -6.7042 55.6577...
5 answers
Suppose a farmer has 1500 feet of fencing to surround arectangular field. If fencing is required around all sides, x isthe length of the fence, and y is the width, write a formula forthe total area enclosed as a function of x.
Suppose a farmer has 1500 feet of fencing to surround a rectangular field. If fencing is required around all sides, x is the length of the fence, and y is the width, write a formula for the total area enclosed as a function of x....
4 answers
~12 points SerPSE1O 5.8.OP.033_ 0/6 Submissions UsedMatesAsk Your Teacherubbetducaon aninclined nlanu When the angle Unciinaton slide down the Incilne. Tinen Wha n (ne Jtlt decreased to J0 the speed statlc and kinatlc frction betwcen the toy duck and the Incling?tne plunu ncnised 35.90 the toy duck beqing toy duci constant; What Je the coefficients ofstulickinetic
~12 points SerPSE1O 5.8.OP.033_ 0/6 Submissions Used Mates Ask Your Teacher ubbet duca on aninclined nlanu When the angle Unciinaton slide down the Incilne. Tinen Wha n (ne Jtlt decreased to J0 the speed statlc and kinatlc frction betwcen the toy duck and the Incling? tne plunu ncnised 35.90 the toy...
5 answers
Lnedealaem nunean olcedeaamac Ei Oach nunbel ol eghi fnelan (anjemk M eciad Lcupht Camcltt PansD 25.971 18 146 715,421 14459 2 = 2467= 1293 16.42 174 12 760 47, 203 16 637 1633Lmnncinct lctel bata dlam IhaJmong coup# Malz {Ceak tanel molcsaciemeiIn "neInncieVleene ddengetaenondlalnn nlalleatecl dala Mtenereachin tidual daftenc~deenedenneterieoerAenecolrAienat hnstee [uaheethed[4?noicitnorc # Metaqaat DeeEnalnoronrt
Lnedealaem nunean olcedeaamac Ei Oach nunbel ol eghi fnelan (anjemk M eciad Lcupht Camcltt Pans D 25.971 18 146 715,421 14459 2 = 2467= 1293 16.42 174 12 760 47, 203 16 637 1633 Lmnncinct lctel bata dlam IhaJmong coup# Malz {Ceak tanel molcs aciemei In "neInncie Vleene ddengetae nondlalnn nlall...
5 answers
Three identica point charges_ each of mass 0.200 kg_ hang from three strings as shown in the figure below. the lengths of the left and right strings are each 24.0 cm, and if the angle is 450_ determine the value of 2.4550-6 Your response within 10% of the correct value This may be due to roundoff error; or voU could have mistake in your calculation_ Carry out all intermediate results to at least four-digit accuracy to minimize roundoff error:
Three identica point charges_ each of mass 0.200 kg_ hang from three strings as shown in the figure below. the lengths of the left and right strings are each 24.0 cm, and if the angle is 450_ determine the value of 2.4550-6 Your response within 10% of the correct value This may be due to roundoff er...
5 answers
1) For the system shown below; if M = 6.0 kg, what is the tension in string 1?30 60Ma) 39 Nb) 34 Nc) 29 Nd) 44 Ne) 51 N
1) For the system shown below; if M = 6.0 kg, what is the tension in string 1? 30 60 M a) 39 N b) 34 N c) 29 N d) 44 N e) 51 N...
5 answers
Lct2 6^ ^ :'bcL ricJ4) C~r-L $5 , + {47 + +nh fin +e~> ' 26) L4t 2 -^ Bc + 5871<J2Sho +-5^Ll-F642 a4riu ~BevL 2 0" 0:i 4 La+ R , +C erot + f9 444 4t TA 6 + 4 p' F 2 B^ Fic4 44 ~Ppet bbu^d for R5 _Usi d T^ Low Anan +r~J 64 2 + 8i 4^ 6t 6 7444 4"' R ^ + 02 4 +^ 000 |
Lct 2 6^ ^ :' bc L ricJ 4) C~r-L $5 , + {47 + +nh fin +e~> ' 2 6) L4t 2 -^ Bc + 5871<J 2 Sho +- 5^ Ll-F 642 a4riu ~BevL 2 0" 0:i 4 La+ R , +C erot + f9 444 4t TA 6 + 4 p' F 2 B^ Fic4 44 ~Ppet bbu^d for R5 _ Usi d T^ Low Anan +r~J 64 2 + 8i 4^ 6t 6 7444 4"' R ^ + 0...
5 answers
Find the inverse Laplace transfon:(s-1)6(1-e s2+9
Find the inverse Laplace transfon: (s-1) 6(1-e s2+9...

-- 0.021079--