Line integral Definition and 404 Threads

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...

Homework Statement http://s2.ipicture.ru/uploads/20120117/ReWSCD1f.jpg The attempt at a solution \frac{\partial P}{\partial y}=\frac{2y}{x^3} \frac{\partial Q}{\partial x}=\frac{2y}{x^3} \frac{\partial Q}{\partial x}=\frac{\partial P}{\partial y} According to my notes: Both functions are...

Homework Statement ∫C1(0) dz / (z * sin2(z)) Homework Equations Residue Theorem material The Attempt at a Solution z * sin2(z) = z * (1/2 - cos(2z)/2) = z * [1/2 - (1/2)∑(-1)n(2z)n/(2n)! ] = z3 + ... ---> z * sin2(z) has a zero of order 3 at z = 0 ---> 1/(z * sin2(z))...

Homework Statement Find the area swpet out by the line from the origin to the ellipse x=cos t and y=sin t as t varies from 0 to t_0 where t_0 is constant between 0 and 2 pi Homework Equations Use A= \frac{1}{2} \oint_{C} y dx -xdy The Attempt at a Solution How does one...

Homework Statement Find ∫(ez+cos(z))/z dz integrated over C1(0) Homework Equations Theorem 6.10 (Cauchy's integral formula) Let f be analytic in the simply connected domain D and let C be a simple closed positively oriented contour that lies in D. If z0 is a point that lies interior to C...

Let \alpha be circle in the complex plane centered at z=1 with radius r=3/2. I proceed by partial fraction decomposition and then use Cauchy's Integral Formula. \int_\alpha \frac{z^7 -1}{z^6 - z^2}dz = \int_\alpha zdz - \int_\alpha \frac{1}{z^2}dz +\frac{i}{2}\int_\alpha \frac{z-i}{z^2 - i}dz...

Homework Statement Evaluate the line integral: int(ydx+xdy) where the path C is y=sinx from (0,0) to (pi/2,0) Homework Equations The Attempt at a Solution (pi/2,0) is not a solution to y=sinx. I could use the fundamental theorem but for my potential function I get F(x,y)=xy...

The question is compute the integral over c of (x/y) where c is the line segment from (2,4) to (5, 25) followed by the parabolic arc from (5, 25) to (10, 100) I tried setting this up in terms of x and then y using the line integral formula but I am got a negative answer which i know can't be...

Homework Statement Let C be the (positively oriented) boundary of the first quadrant of the unit disk. Use the definition of the line integral to find ∫(xy)dx+(x+y)dy Homework Equations x=rcos(x) y=rsin(x) dx=-sin(x) dy=cos(y) 0≤ t ≤ ∏/2 The Attempt at a Solution...

∫Homework Statement Use the definition to find the line integral of F(x,y) = (y,x) along each of the following paths. The parabola y = x^2 from (-1,1) to (1,1) Homework Equations F(x) = gradientf(x) ∫F(x) dx = f(b) - f(a) The Attempt at a Solution I tried (y,x) dot...

Homework Statement Find the line integral of ∫ x+yz dx + 2x dy + xyz dz C consists of line segments from (1,0,1) to (2,3,1) and from (2,3,1) to (2,5,2). Homework Equations r=(1-t)<r0> + t<r1> 0<t<1 The Attempt at a Solution I split up the two line segments into C1 and C2...

Homework Statement ∫dz/(z4+1) integrated over the curve C1(1+i) Homework Equations The only thing we learned in this chapter is Cauchy's integral formula, so I'm assuming that comes in somehow. The Attempt at a Solution ∫dz/(z4+1) = ∫dz/(z+1+i)(z+1-i)(z-1+i)(z-1-i) Not bad...

Homework Statement Find the mass and the coordinates for the center of mass of a thin wire formed like a quarter circle. Homework Equations Circle equation: x2+y2=r2 Mass density: rho=x+y The Attempt at a Solution I know that: x2+y2=cos2(t)+sin2(t)=1 This leads to...

Homework Statement Using a partial fraction decomposition, show that if z lies in the right half plane and C is the line segment from 0 to z, then ∫C dz/(z2+1) = i/2 Log(z+i) - i/2 Log(z-i) + π/2 Homework Equations Log(z) = ln(z) + i Arg(z) (maybe relevant?) The Attempt at...

Homework Statement f(x,y) = \sqrt{1+9xy}, y = x^{3} for 0≤x≤1 Homework Equations The Attempt at a Solution I don't even know how to start this problem. I thought about c(t) since that's all I have been doing, but there isn't even c(t). I only recognize domain. Can anyone help me...

Here is the question: A very thin wire which follows a semicircular curve C of radius R,lies in the upper half of the x-y plane with its center atthe origin. There is a constant current I flowing counter clockwise, starting upward from the end of the wire on the positive x-axis and ending...

Homework Statement \displaystyle \int x^2dx+y^2dy+z^2dz where C is the line segment from(0,0,0) to (1,2,-1) and (1,2,-1) to (3,2,0) Homework Equations \displaystyle\int_c \vec F(t) d \vec r(t)= (x^2 i+ y^2 j+z^2 k)d \vec r(t) where d \vec r(t) for C_1=ti+2tj-tk and d \vec r(t) for...

Homework Statement \displaystyle \int_c zdx+xdy+ydz where C is given by t^2\vec i +t^3 \vec j +t^2 \vec k Can this \displaystyle \int_c zdx+xdy+ydz be written as \displaystyle \int_c z\vec i+x \vec j+y \vec k? I believe I need to evalute the integral \displaystyle \int_c \vec F(...

Homework Statement \int_{c}cos (x)dx+sin(y)dy where c consist of the top half of the circle x^2+y^2=1 from (1,0) to (-1,0) The Attempt at a Solution Do I parameterise x=t and then y becomes y= (1-t^2)^{1/2}...? Replace the corresponding dx and dy and then integrate between the limits?

Homework Statement \int_{C}|y|ds where C is the curve (x^{2}+y^{2})^{2}=2^{2}(x^{2}-y^{2}) Homework Equations The Attempt at a Solution i used polar coordinates x = r cos \theta and y = r sin \theta then substituted into the equation to get r = 2\sqrt{cos 2\theta} since r\geq0 gives...

Hi experts what is line integral - for example if I can draw graph of parabola and i can calculate the area under the graph. But how can i measure the length of parabolic path.

Find the area of the surface consisting of the part of the sphere of radius 2 centered at origin that lies above the horizontal plane z = 1. (Equation of this sphere is given by x^2 + y^2 + z^2 = 2^2 .) x^2+y^2+1=4 x^2+y^2=3 This is the base of the solid. But how do we find the required...

Homework Statement The Problem states: Given the force vector field(in polar coordinates) : F(r,\theta)=-4Sin\thetai+4Sin\theta j, compute the work done in moving a particle from (1,0) to the origin along the curve whose polar equation is : r=e^{-\theta} The Attempt at a Solution I...

How do you work out the parameterization for a line integral. I have this example, and the closed curve C bounds the lines y=0, x=2 and y^2 = 8x. In the solution to the problem it states that there are many parameterizations available. So I just wanted to know, how do you work out the...

find the value, \int\limits_{0}^{2\Pi} e^{-\sin t} \sin\lbrace (\cos t ) - (n-1) t \rbrace dt ? I have no idea...

I can't find the value, for natural number n = 1, 2, 3, ... I = \int\limits_{C}\dfrac{e^{iz}}{z^n} dz find the value. where z(t) =e^{it} , 0\leq t \leq 2\Pi

Homework Statement calculate the integral f · dr for the given vector field f(x, y) and curve C: f(x, y) = (x^2 + y^2) i; C : x = 2 + cos t, y = sin t, 0 ≤ t ≤ 2π (2pi) Homework Equations Would the vector F simply be <(x^2+y^2),0> since there is no j component? The solution is 4pi...

I'm not getting the answer from the back of the book for some reason. Is the book wrong or am I wrong? Homework Statement calculate \intf · dr for the given vector field f(x, y) and curve C: f(x, y) = (x^2 + y^2) i; C : x = 2 + cos t, y = sin t, 0 ≤ t ≤ 2πHomework Equations itex]\int[/itex]f ·...

Homework Statement integral (sin^4(x) + cos^4(x))^.5 dx Homework Equations sin^2(x) = (1 - cos^2(x)) The Attempt at a Solution cos^4(x) + {1 - cos^2(x)}^2 = 2cos^4(x) -2cos^2(x) + 1 subst u = cos^2(x) integral (2u^2 - 2u +1)^.5

There are line integral with respect to arc length and line integral with respect x/y. I know \int_C Pdx+Qdy is useful to calculate the work. When do we need the line integral with respect to arc length?

Homework Statement \int_{C} (xy^{2}-3y)dx + x^{2}y dy G is finite region enclosed by: y=x^{2} y=4 C is boundary curve of G. Verify Green's Theorem by evaluating double integral and line integral. The attempt at a solution Q = x^{2}y dQ/dx = 2xy P = xy^{2}-3y dP/dy =...

Bachman's "line integral" versus "classical line integral" David Bachman A Geometric Approach to Differential Forms http://arxiv.org/abs/math/0306194 When Bachman talks, in Appendix A, about "classical" line, surface, volume integrals, does he mean integrals of differential 0-forms (scalar...

Homework Statement ∫(zdx+xdy+ydz) along the curve C: x(t)= cos(t), y(t)= sin(t), z = 3t, Boundaries are 0 and 2pi Homework Equations General integration and differentation. The Attempt at a Solution given the values I calculated that: Using chain rule: dx = -sintdt...

Homework Statement F = ( 2y i + 3x J + z2 k where S is the upper half of the sphere x2 + y2 + z2 = 9 and C is its boundary. Homework Equations The Attempt at a Solution I used Stoke's Theorem and found the solution to be 36 pi, but when I use line integral to verify, using...

Homework Statement http://img534.imageshack.us/img534/6859/unledei.jpg 3. The Solution [PLAIN][PLAIN]http://img607.imageshack.us/img607/3104/unledfe.jpg Why did they switch the order of x - 2xy3 in Green's Theorem?

I figure it out already.

Homework Statement Two circular cylinders of radius a intersect so that their axes meet at right angles. Use a line integral to find the area of the part from one cut off by the other. Homework Equations line integral formula The Attempt at a Solution I'm lost as to where to set...

Homework Statement Find the work done by the force field F(x,y) = x sin(y)i + yj on a particle that moves along on the parabola y = x^2 from (-1,1) to (2,4). Homework Equations Work = line integral of the dot product of Field vector and change in the path The path is parabola equation...

Homework Statement Find the mass and center of mass of a wire in the shape of the helix x=t, y=\cos{t}, z = \sin{t}, 0 \le t \le 2 \pi, if the density at any point is equal to the square of the distance from the origin. Homework Equations Arc length formula: ds =...

I have to integrate |z|2dz from 0 to 1 + 2i using the indicated paths. The first path is a straight line from the origin to 1 + 2i and the second has two lines, the first going from 0 to 2i along the y-axis and then from 2i to 1 + 2i, a line parallel to the x axis. For the first path, the...

I have to evaluate this line integral in the complex plane by direct integration, not using Cauchy's integral theorems, although if I see if a theorem applies, I can use it to check. \int (z^2 - z) dz between i + 1 and 0 a) along the line y=x b) along the broken line x=0 from 0 to 1...

Hi guys, can anyone help me with evaluating this: \int^{}_C |y| \,ds where C is the curve (x^2+y^2)^2=r^2(x^2-y^2) any hints with the parametrization?

Homework Statement A wire lies along the piecewise linear curve extending from the point (2,2) to the point (12,4) to the point (12,9). If the density of the wire is given by (xy)=2xy+6x, use a line integral to find the mass of the wire. Homework Equations The Attempt at a Solution...

Homework Statement Evaluate the line integral along the segment P1(0,3) to P2(-3,0) of the circular path shown in figure. Figure basically shows a circle with a radius of 3. The part that i have to evaluate is from the y-axis (P1) to the x-axis (P2), basically a quarter of the circle...

Homework Statement Compute the Work of the following line integrals in the vector field \vec{V}=(2x^{2}-3y;4xy;3x^{2}z) Homework Equations For the following lines: Curve1: \vec{r}(a)=(a,a,a^{2}); \ 0\le a \le 1 Curve2: \vec{r}(a)=(a,a^{2},a); \ 0\le a \le 1 The Attempt at a Solution...

The question is suppose that F is an inverse square force field, that is, F(r)=cr/|r^3| where c is some constant. r = xi + yj + zk. Find the work done by F in moving an object from a point P1 along a path to a point P2 in terms of the distances d1 and d2 from those points to the origin...

Homework Statement Consider a curve in R2 given in polar coordinates r=r(θ) for θ1<= θ <= θ2. Show that the line integral is equal to the integral from θ1 to θ2 of f(r*cosθ, r*sinθ) sqrt (r^2 + (dr/dθ)^2) dθ Homework Equations x= cos θ, y= sin θ The Attempt at a...