Line integral Definition and 393 Threads

Suppose I have a vector V and I want to compute for the line integral from point (1,1,0) to point (2,2,0) and I take the path of the least distance (one that traces the identity function). The line integral is of the form: \int _a ^b \vec{V} \cdot d\vec{l} Where: x=y, \ d\vec{l}...

Homework Statement Evalutate \int_{C} (y2-3x2)dx + (2xy+2)dy, where C is a smooth curve from (0,1) to (1,3). 2. The attempt at a solution I've checked through my notes and text but can't find an example. I'd appreciate it if someone could help me get this started.

I'm trying to solve this integral as x-> Infinity \int \frac{dz}{8i + z^2} ...on the y=x line, but I have no idea what I'm doing. The book I'm using is less than helpful in this regard. I'm not supposed to use any complex analysis tools (Cauchy, etc), but just solve it as a line...

Homework Statement Evaluate the line integral ∫(x+2y)dx+(x^2)dy, where C consists of the line segments from (0,0) to (2,1) and (2,1) to (3,0) Homework Equations The Attempt at a Solution I'm unsure of what to do. I did (1-t)r0 + t(r1) for (0,0) to (2,1) and (2,1) to (3,0). I...

Homework Statement Green's Theorem to evaluate the line following line integral, oriented clockwise. ∫xydx+(x^2+x)dy, where C is the path though points (-1,0);(1,0);(0,1) Homework Equations Geen's theorem: ∫F°DS=∫∫ \frac{F_2}{δx}-\frac{F_1}{δy} The Attempt at a Solution What...

I've attached the problem as a picture. Generally, this would be a simple problem if I were to apply Green's theorem. But I can't use Green because D would not be a simply connected closed region; the vector field isn't defined at the origin. Could someone please give me an idea of where to...

Homework Statement \int_C \mathbf F\cdot d \mathbf r where \mathbf F = x^2\vec{i}+e^{\sin^4{y}}\vec{j} and C is the segment of y=x^2 from (-1,1) to (1,1). Homework Equations \int_C \mathbf F\cdot d \mathbf r=\int_a^b \mathbf F( \mathbf r(t))\cdot r'(t) dt=\int_C Pdx+Qdy where \mathbf F =...

Evaluate the line integral, where C is the given curve. \int_{c} xy\:ds, when C: x=t^{2}, \ y=2t\ , \ 0\leq t\leq4 To solve this I should use the formula \int^{b}_{a} f(x(t),y(t))\sqrt{(\frac{dx}{dt})^{2}+(\frac{dy}{dt})^{2}}dt This gives me \int^{4}_{0}...

So I'm able to calculate them no problem. But the problem is, I don't really understand what they mean. For example, W=∫Fdr I understand that a vector field is something that defines a vector at every point. Then if we pick a curve in the field, we are integrating along that curve. So does...

Given value of a line integral, find line integral along "different" curves Homework Statement I think I've got this figured out, so I'm just checking my answers: Suppose that \int_\gamma \vec{F}(\vec{r}) \cdot d\vec{r} = 17 , where \gamma is the oriented curve \vec{r}(t) = \cos{t} \vec{i}...

Homework Statement A squirrel weighing 1.2 pounds climbed a cylindrical tree by following the helical path x = \cos{t}, y = \sin{t}, z = 4t, 0 \leq t \leq 8 \pi (distance measured in feet) How much work did it do? Homework Equations \int_{C} \vec{F} \cdot d\vec{r} The Attempt...

Hi I'd like to know who is the mathematician or physicist that create line integral, I've always studied how calculate it, but Who can that be, I wonder? Thanks Alfonso.

I met a proof problem that is as follows. ##\bf a = ∫_S d \bf a##, where S is the surface and ##\bf a ##is the vector area of it. Please proof that ##\bf a = \frac{1}2\oint \! \bf r \times d\bf l##, where integration is around the boundary line. Any help would be very appreciated!

Hi, I'm trying to calculate some line integral with both Gauss' and Stokes' theorems, but for some strange reasons I get different results. Since the solution with Stokes' theorem seems to be somewhat easy I doubt that this question was meant to be solved by Gauss' theorem but I still want to...

Homework Statement Find the value of the line integral \int C(e-x^3 - 3y)dx + (tan y + y4 + x) dy where C is the counterclockwise-oriented circle of radius 4 centered at (0,2). The Attempt at a Solution First off, I didn't think this was path independent since the derivative of the dx term...

Analyzing an integral over a non-exact region for gamma defined by |x|+|y|=4 The following was similar to a problem on a calculus final that I got wrong. It is an extension of a problem in R.C. Buck "Advanced Calculus" on page 501. Similar to knowing the trick to integrating e^{|x|} (which...

I have the vector: {\bf{u}}(x,y) = \frac{{x{\bf{i}} + y{\bf{j}}}}{{{x^2} + {y^2}}} Where: x = a\cos t y = a\sin t I know I need to use the equation \int\limits_0^{2\pi } {{\bf{u}} \cdot d{\bf{r}}} And the answer is \int\limits_0^{2\pi } {} ((a\cos t/{a^2})( - a\sin t) +...

Homework Statement Evaluate ∫(x^3 + y^3)ds where C : r(t)=<e^t , e^(-t)>, 0 <= t <= ln2 c Homework Equations The Attempt at a Solution I tried to parametrize the integral and change ds to sqrt(e^(2t) + e^(-2t)) dt. I then change (x^3 + y^3) to...

I have no idea how to even start with this problem. I know the basics but this one just gets complicated. Please guide me! Find the line integral: ∫C {(-x^2 + y^2)dx + xydy} When 0≤t≤1 for the curved line C, x(t)=t, y(t)=t^2 and when 1≤t≤2, x(t)= 2 - t , y(t) = 2-t. Use x(t) and...

I have no idea how to even start with this problem. I know the basics but this one just gets complicated. Please guide me!Find the line integral: ∫C {(-x^2 + y^2)dx + xydy} When 0≤t≤1 for the curved line C, x(t)=t, y(t)=t^2 and when 1≤t≤2, x(t)= 2 - t , y(t) = 2-t. Use x(t) and y(t) and...

Homework Statement Calculate \oint _c xdy-ydx where C is the straight line segment from (x_1 , y_1) to (x_2 , y_2) Homework Equations The Attempt at a Solution so from Greens theorem I get \oint _c xdy-ydx = \int\int \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} dxdy =2...

Consider that the curve C (IR3) is the intersection between x2+2z2=2 and y=1. Calculate the line integral: Note: The curve C is traversed one time in the counter-clockwise direction (seen from the origin of IR3). My attempt: http://i.imgur.com/5EsJO.png Can someone check? Thanks! :)

Homework Statement I have exam tomorrow and there's a problem I don't know how to do. Consider the curve C1 (x=-y^2+3y) and C2 (x=0), both defined for y\in[0,3]. Calculate the work done by F(x,y)=(x,y^2) along the curve C=C1UC2 (retrograde direction). Homework Equations The...

line integral -- confusion on squares and square root terms Homework Statement Do you see where they have sqrt(16 sin^2t etc = 5? How do they get that, the answer should be 7, the square root of 16 is 4, sin^2 + cos^2 is 1 and the square root of 9 is 3, 3 + 4 = 7. It's like they're...

I have shown by my intuition that if a good field g(2th or more differentiable) in n dimension satisfies \frac{∂g_{i}}{∂x_{j}}-\frac{∂g_{j}}{∂x_{i}}=0 for all i,j, then \ointg\cdotdl=0, hence there exist a scalar function \phi such that \frac{∂\phi}{∂x_{i}}=g_{i} for all i. I want to...

Homework Statement I have attached the problem to the post.Homework Equations Properties of line integral. Path independence. The Attempt at a Solution I have shown that the path is independent, as: \partial P/\partial y = \partial Q/\partial x The problem is with the parametrization. I found...

Homework Statement a) Evaluate the work done by the force field F(x, y) = (3y^(2) + x)i + 4x^(3)j over the curve r(t) = e^(t)i + e^(3t)j, tε[0, ln(2)]. b) Using Green’s theorem, find the area enclosed by the curve r(t) and the segment that joins the points (1, 1) and (2, 8). c) Find the...

In the line integral of work done by conservative forces, should we care about the direction of force and differential displacement if we choose different coordinate system? The formula of potential energy comes out to be different if we choose different coordinate system.

Homework Statement -- Homework Equations -- The Attempt at a Solution This isn't really a proper homework question so I'll just write my problem here: I'm trying to compute a closed line integral over a triangular region. I have calculated two of the sides, but am now left...

Homework Statement Evaluate the line integral over the curve C \int_{C}^{}e^xdx where C is the arc of the curve x=y^3 from (-1,-1) to (1,1) Homework Equations \int_{C}^{}f(x,y)ds=\int_{a}^{b}f(x(t),y(t))\sqrt((\frac{dx}{dt})^2+(\frac{dy}{dt})^2)dt The Attempt at a Solution I tried...

I am a little confused about how to generally go about applying Stokes's Theorem to cylinders, in order to calculate a line integral. If, for example you have a cylinder whose height is about the z axis, I get perfectly well how to parameterize the x and y components, using polar coordinates...

Let $${\tmmathbf\vec{F} = yx^2 \tmmathbf\hat{i} + \sin (\pi y) \tmmathbf\hat{j}}$$, and let $${C}$$ be the curve along the line segment starting at (0,2) and ending at (1,4). $${\int_C \tmmathbf\vec{F} \cdot d \tmmathbf\vec{r} =}$$ My path comes out to be: r=ti +2tj dr=i+2j...

If $${\tmmathbf{r}}$$ and $${\tmmathbf{s}}$$ are piecewise smooth paths, which have the same graph, then they are said to be equivalent paths. They either trace out a set of points in the same direction, or in the opposite direction. If they trace out a curve $${C}$$ in the same...

If $${C}$$ is the straight line that connects points in the plane $${(x_i, y_i)}$$ and $${(x_f, y_f)}$$, find a path $${\tmmathbf{r} (t)}$$ that traces out $${C}$$ starting at the initial point $${(x_i, y_i)}$$ and ending at $${(x_f, y_f)}$$ as $${t}$$ goes from zero to 1. Now the path that...

I read(and numerically calculated) that the line integral around a circle of radius r centered at 0 for the function conjugate(z)/(z - t) is 0 for all t inside the circle. I don't know see how this integration is performed.

Homework Statement Observe the curve C, which is given as the intersection between the surfaces: x2+z2=1 and y = x2 Calculate \int_c{\sqrt{1+4x^2 z^2}} Homework Equations So basicly I'm completely lost on this assignment. So far I can see, that x^2+z^2=1 describes a...

Hi. I have a concrete doubt with this problem. Here's the pic. It asks me to calculate the work done by force P (the ball moves with constant speed). So the solution is in the book and I understood everything, but the problem comes here, the force P in axis y is zero so the work of P should...

Homework Statement http://img39.imageshack.us/img39/6669/wileyplus.png The field is conservative. With their description of C, and using the Fundamental Theorem of Calculus for Line Integrals, would you evaluate f(11/sqrt(2), 11/sqrt(2)) - f(0,0) or f(11/sqrt(2)...

Homework Statement Calculate the line integral: f(x,y) = (x² - 2xy)î + (y² - 2xy)j, between the points (-1,1) and (1,1) along the parabola y = x². (resp: -14/15) The attempt at a solution I thought something like this: substitue y = x², and then integrate de f(x,y). And then evaluate...

Homework Statement Evaluate \int3x^2dx+2xydy, where C is the curve x^2+y^2=4 starting at (2,0) and ending at (0,2) in the anti-clockwise direction. The attempt at a solution The curve C is a quarter-circle with centre (0,0) and radius=2. Making y subject of formula: y=+\sqrt{4-x^2} since the...

Hi, can someone tell me where I can find the term: "homeotopic" (or, something like that, I don't know how to write in English). My professor mentioned that term in the line integral, here it is: Let \Omega \subseteq \mathbb{R}^k be area (open and connected set). Curves \varphi, \psi...

Homework Statement To evaluate the following line integral where the curve C is given by the boundary of the square 0 < x < 2 and 0 < y < 2 (In the anti clockwise sense): \oint (x+y)^2 dx + (x-y)^2 dy The Attempt at a Solution Firstly it is noted that for a square ABDE : Between...

Homework Statement Trying to evaluate the following line integral: integral F ds where F = (6(x^2)(y^2), 4(x^3)(y) + 5y^4) and the path is the boundary curve of the first quadrant below y = 1-x^2 in a clockwise direction. Homework Equations The Attempt at a Solution So since...

Homework Statement Find the parameterized curve gamma and vector field F so that the \int\gamma F ds = \int\int2xy dx dy by Green's Theorem. where -2<y<2 1-sqrt(4-y2) < x < 1+sqrt(4-y2) The Attempt at a Solutionx = 1 + sqrt(4-y2) (x-1)2=4-y2 (x-1)2+y2=4 so the path is a circle centered at...

Homework Statement Evaluate the following line integral ∫y^2 dx + x dy where C is the line segment joining the points (-5,-3) to (0,2) and is the arc of the parabola x= 4-y^2 Homework Equations Green's Theorem ∫ Mdx + Ndy = ∫∫ (∂N/∂x - ∂M/∂y ) dy dx The Attempt at a...

Okay this might be a nooby question, but it bothers me. What is the difference between the line integral of a scalar field and just a regular integral over the scalar field? For a function of one variable i certainly can't see the difference. But then I thought they might be identical in...

Homework Statement Given the vector field \vec{v} = (-y\hat{x} + x\hat{y})/(x^2+y^2) Show that \oint \vec{dl}\cdot\vec{v} = 2\pi\oint dl for any closed path, where dl is the line integral around the path.Homework Equations Stokes' Theorem: \oint_{\delta R} \vec{dl}\cdot\vec{v} = \int_R...

Homework Statement Evaluate the line integral ∫c y2 dx + 2xy dy, where C, is the path from (1, 2) to (2, 4) parametrised by r(t) = (t2 + 1)i + (2t2 + 2)j , 0 ≤ t ≤ 1 Homework Equations I worked out the velocity magnitude |v(t)| as 2t√5 The Attempt at a Solution I simply integrated...

Homework Statement http://s2.ipicture.ru/uploads/20120204/iuPLuS1l.png The attempt at a solution x=t^2-1 and y=t^2+1 \frac{dx}{dt}=2t and \frac{dy}{dt}=2t The line integral is of the form: \int P\,.dx+Q\,.dy So, i use direct substitution: \int^1_0 4t^3-2t\sin(t^2-1)+2t\cos(t^2+1)\,.dt Is...

Hi, I am trying to get a useful heuristic picture of a line integral, like the area under a curve for an ordinary integral. My current one is: if I place a particle in a force field, then the line integral from point A to B is the change in kinetic energy of the particle from A to B. This...