Line integral Definition and 404 Threads

Homework Statement See figure attached for problem statement as well as the solution provided. Homework Equations The Attempt at a Solution My only question is why can't I use, 32 \int ^{1}_{2} t\sqrt{8t^{2}+1}dt I realize that this will give me the negative of his answer...

Homework Statement Evaluate the line integral, \int_{C}x^{2}yzds, \text{ where C is the curve } z = x + y, \quad x + y + z = 1\text{ from }(1,\frac{-1}{2},\frac{1}{2}) \text{ to } (-3, \frac{7}{2}, \frac{1}{2}) Homework Equations The Attempt at a Solution Here's my attempt at...

Hi All, I have been battling with this question for a while. Given a conservative vector field, we know that there are infinitely many closed paths where the line integral evaluated is zero. In fact this is the requirement for a conservative vector field: Every line integral of any closed...

I have the parametrization of C x=9cos(t) y=9sin(t) z=-8t 0<=t<=10*pi and I have to calculate \int_c 7x^2+4y^2 -5xy after I transform this I get (7*81cos^2(t) +4*81sin^2(t) -5*81*cos(t)*sin(t))*sqrt(-9sin^2(t)+9cos^2(t)-8) dt from 0 to 10*pi Now that is a monster. What did I do wrong?

Homework Statement Use Your Phi(from part 1) and the fundamental theorem of line integrals to evaluate the same line integral. (should get the same answer!) The Attempt at a Solution Phi from part 1: Phi = xy+ y^2 +C The line from before go from (0,2) to (-2,0) r(a) = (0,2)...

Homework Statement a. Find a parametric equation to describe a parabola from the point (1,1) to the point (2,4). b. Evaluate the line integral \int_C x ds along the parabolic segment in part a. Homework Equations \int_C x ds = \int_{t1}^{t2} x(t) |r'(t)| dt The Attempt at a Solution Well...

I am agonizing about the following integral identity: \frac{d}{dt} \int \int_{g(x,y) \leq t} f(x,y) dx dy = \int_{g(x,y)=t} f(x,y) \frac{1}{\left| \nabla g(x,y) \right|} ds, where ds is the line element. Clearly, using the Heavisite step function, the condition g(x,y) \leq t is...

Homework Statement Without parameterizing the path, determine what the value of the line integral (integral of F dot dr) is, if C is the closed, oriented path that travels around the triangle with vertices (0,0) (5,2), and (-3,6) and F=yi + xj Homework Equations Curl possiblY? The...

Solved Thanks

Homework Statement for \varphi(x,y)=2x+y+10 ,calculate the flux line integral...on a straight line from A(1,4) to B(5,1). Homework Equations The Attempt at a Solution I tried to solve it but didnt get the right answer. first i found the quation of the line which i found to be...

Homework Statement A table of values of a function f with continuous gradient is given. Find the line integral over C of "gradient F dr" where C has parametric equations x = t2 + 1, y = t3 + t, 0<=t<= 1. Sorry, don't know latex. But here's a picture of the table and values...

http://img2.imageshack.us/img2/5061/14983795.jpg I have no idea how they simplified the integral to the second step.

Homework Statement Let f(x,y) and g(x,y) be continuously differentiable real-valued functions in a region R. Show that ∫f ∇g · dr ]= − ∫g ∇f · dr for any closed curve C in R. Homework Equations The Attempt at a Solution I don't really know where to start, so I tried to evaluate...

Homework Statement Evaluate the following line integral on the indicated curve C \int(y^2-x^2)ds C: x = 3t(1+t), y=t^3 ; 0 <= t <= 2 Homework Equations ds = \sqrt{(f'(t))^2+(g'(t))^2}dt The Attempt at a Solution dx/dt = 3+6t dy/dt = 3t^2 ds = \sqrt{(3+6t)^2+(3t^2)^2}dt ds =...

Hello, I am attempting to calculate the line integral of the vector field Line integral of a vector field \overline{A}= x^{2} \hat{i} + x y^{2} \hat{j} around a circle of radius R ( x^{2} + y^{2} = R^{2} ) using cartesian coordinates. The appropriate differential line element in cartesian...

I am attempting to calculate the line integral of the vector field \overline{A}= x^{2} \hat{i} + x y^{2} \hat{j} around a circle of radius R (x^{2} + y^{2} = R^{2}) using cylindrical coordinates. It is simple enough to convert the x and y components to their cylindrical counterparts, but I am...

Homework Statement You are given a vector field A= kx2 x. a. First, calculate the line integral of A from x=-2 to x=2 along the x axis. b. Next, calculate the line integral of A between the same 2 points, but use a semicircular path with a center at the origin. Recall that in cylindrical...

So this is an ultra basic question, but I'm rusty with parametrization techniques and wanted to make sure I was doing this correctly. Let's say I want to evaluate \int_{\gamma} z \: dz where \gamma : [a,b]\rightarrow \mathbb{C} is some path of integration. Now, I figure I can parametrize the...

Homework Statement \int_{C}(xy+\ln{x})\mathrm{d}s where C is the arc of the parabola y=x^2 from (1,1) to (3,9) Homework Equations \int_{C}f(x,y)\mathrm{d}s = \int_{a}^{b}f(x(t),y(t))\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}\mathrm{d}t The Attempt at a Solution Ok, so...

Homework Statement \ointxydx+x^2dy C is the rectangle with vertices (0,0),(0,1),(3,0), and (3,1) Evaluate the integral by two methods: (a) directly and (b) using green's theorem. Homework EquationsThe Attempt at a Solution Evaluating the integral directly: c1: y=0,x=t,dx=dt,dy=o...

The standard method of calculating area of ellipse: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 Area = \int_C -ydx \hbox { or } \int_C xdy It is more convient to use polar coordinate x=a cos \theta \; \hbox { and }\; y=b sin \theta dy = b cos \theta \hbox{ Using } \int_C xdy =...

Homework Statement [PLAIN]http://img843.imageshack.us/img843/3995/calc3.jpg The Attempt at a Solution Im here asking for some help in direction on how to do these problems so that I can find the solution by myself... I would really really appreciate any help anyone could provide...

Homework Statement hello again, sorry for asking so many questions, i just want to make sure if I am correct or not calculate the line integral y^2dx+x^2dy where line C is the triangle with sides x=1, y=0 and y=x The Attempt at a Solution first of all i tried to find a customization of the...

My course notes said that in greens theorem where the closed line integral of F.r = the double integral (...)dxdy the curve c is taken once anti-clockwise, why does it matter which way you take the line integral? Does it matter at all? Thanks

Homework Statement I'm attempting Q 3 from ch 16.4 of Stewart (p 1060). We are required to find the line integral where C is the triangle with vertices (0,0), (1,0) and (1,2). The line integral is Int xy dx + x^2*y^3 dy Homework Equations The Attempt at a Solution...

Homework Statement Calcualte the value of \int\limits_L \sqrt{x^2+y^2}dl, where L is an arc of a logarithmic spiral r=ae^{m\phi} between points A(0,a) and B(-\infty,0). Problem: I can't find a value of \phi where x=-\infty or y=a. Homework Equations We parametrise and get...

Homework Statement Calculate the line integral \int\limits_{AB} x^2 dx+ \sqrt{xy}dy , where AB is a part of a circle in the first quarter of carthesian coordinates system ranging from A(0,R) to b(R,0). Homework Equations The Attempt at a Solution Parametrisation of a circle...

The Integral I is defined by I = Integral F . dr Where F = (x-y, xy) << This is a verticle vector, i just didn'nt know how to write it with latex. And C is a triangle with the vertices (0,0), (1,0) and (1,3) tracked anticlockwise. Calculate the line integral using greens...

Homework Statement Integrate F(x) = x / |x|^3 along the straight line from (1,0,0) to (2,-2,1). Homework Equations Line integral = int (F dot dx) The Attempt at a Solution I don't know where to start. Usually I do line integrals by parameterizing the line and the vector field...

Can line integral of a vector field ever be zero? If can, what is the interpretation of this value (0) ? Thanks.

Hi everyone. I am going through examples for maths exams and am unsure on the final part of a question I am attempting so hoping you may help me? Homework Statement "Let C be the closed, piecewise smooth curve comprising individual curves C1 and C2 defined by r1 = (x, x2, 1) and r2 =...

I want to check my understanding of the line integral: For a scalar line integral, what we have geometrically is the area between a curve a given function, yes? Hence, it can be thought of as a kind of thin wall, correct? And where our function is f(x,y)=1, we have the length of the...

Apostol page 386, problem 5 Homework Statement Given f,g continuously differentiable on open connected S in the plane, show \oint_C{f\nabla g\cdot d\alpha}=-\oint_C{g\nabla f\cdot d\alpha} for any piecewise Jordan curve C. Homework Equations 1. Green's Theorem 2. \frac{\partial...

So, my multivariable class has just started line integrals, and I could use a little help with them. The problem I'm currently working on says: Evaluate the line integral, where C is the given curve: \int\limits_C \! xy \,ds C: x=t^2, y=2t, 0 \le t \le 1 I realize that, by eliminating the...

Homework Statement Compute the line integral of \intc ydx +zdy + xdz where c is the intersection of x^2 +y^2+z^2= 2(x+y) and x+y=2 (in the direction clockwise as viewed from the origin) Homework Equations The Attempt at a Solution While attempting this problem I had a few...

what is the different between line integral and surface integral? If we parameterize curve by x=t , y=t , what is the range of t ? Is it 0<= t <=1? why?

Homework Statement Homework Equations Given above. The Attempt at a Solution I attempted this problem first without looking at the hint. I've defined F(r) as (B+A)/2 + t(B-A)/2, with dr as (B-A)/2 dt . Thus F(r)dr = ((B+A)/2)*((B-A)/2)+((B-A)/2)^2 dt When I integrate this from -1 to 1 I...

Homework Statement Air is flowing with a speed of 0.4m/s in the direction of the vector (-1, -1, 1). Calculate the volume of air flowing per second through the loop which consists of straight lines joining, in turn, the following (1,1,0), (1,0,0), (0,0,0), (0,1,1), (1,1,1) and (1,1,0)...

Evaluate the line integral Force field is the integral in the form of integrand ( (2x dx+ 2y dy + 2 zdz)/r^2). the domain of integral is C, C = C1 + C2. C1 is the line segment from (1; 2; 5) to (2; 3; 3). C2,arc of the circle with radius 2 and centre (2; 3; 1) in the plane x = 2. The...

Homework Statement Evaluate \int[(3x-y)dx-xdy] where C consist of the parabola y=x^2 from (0,0) to (1,1) and then the line segment from (1,1) to (0,1) Homework Equations The Attempt at a Solution i did the integral of the y=x^2 parametrized x=t y=t^2 from 0 to 1 then i got my 1/2 but for the...

In some books I have seen: \oint \mathbf{E} \cdot d\mathbf{s}=0 Since the Electric Field is meant to be conservative. Elsewhere, however, I have also seen: \oint \mathbf{E} \cdot d\mathbf{s} = -\frac{d\Phi_B}{dt} What's going on here? Thanks

Can I consider the ordinary integral over the real line a special case of the line integral, where the line is straight and the field is defined only along the line?

In a line integral with respect to arc length, we have something like f(x, y)ds "inside" the integral sign. The ds tells us that we are working with the arc length function s, taking diferences (s_K+1 - s_k) in the sums that tend to the line integral. Question: do we shall understand that...

Homework Statement What is \int_{\gamma} xy dx + x^2 dy in each of the following cases? 1. \gamma is the lower half of the curve 2x^2 + 3y^2 = 8, traveled from (2,0) to (-2,0). 2. \gamma is the full curve 2x^2 + 3y^2 = 8, traveled counterclockwise. Homework Equations The line...

Hello, sorry for my English;D Homework Statement Can a vector field exist in polar/spherical system? is it possible to define line integral in these systems? does it make any sense a vector field defined in polar system, ex. \vec A\left(r,\varphi\right)=r^3? and a line integral from...

Homework Statement Prove that 2A=\oint \vec{r}\times d\vec{r} Homework Equations The Attempt at a Solution From stokes theorem we have \oint d\vec{r}\times \vec{r}=\int _{s}(d\vec{s}\times \nabla)\times \vec{r}= \int _{s}(2ds\frac{\partial f}{\partial x},-ds+ds\frac{\partial...

suppose i have a nonconservative vector field. and there is a path going from point A to point B. How do i determine the path taken from A to B such that the line integral is maximized? edit: actually after thinkin about it, this might be an undefined problem unless there is some constraint on...

Homework Statement Suppose C is the line segment from the point (1,0) to the point (3,1). Compute the line integral intC {( xdx + (x + y)}dy Homework Equations The Attempt at a Solution i graphed the line that connects(1,0) to (1,3) and i got the equation of that line so y...

Homework Statement What is the result of this? \ointr.dr=? Homework Equations The Attempt at a Solution \ointr.dr =\ointrdr=\int^{a}_{a}rdr = \frac{r^2}{2} \left| ^{a}_{a} = 0 Is it correct?

Homework Statement This is an example in my book, and this is the information in the question. Find the work done by thr force field F(x,y) = (1/2)xy[B] i + (1/4)x^2 j (with i and j vectors) on a particle that moves from (0,0) to (1,1) along each path (graph shows a x=y^2 curve from (0,0)...