Line integral Definition and 404 Threads

Homework Statement Evaluate the line integral by two methods: (a) directly and (b) using Green's Theorem. \oint_C {xy^2 \,dx\, + \,x^3 \,dy} . C is the rectangle with vertices (0,0), (2,0), (2,3) and (0,3). The Attempt at a Solution I get the correct answer using Green's theorem. But I...

1. Evaluate the line integral∫F . dr with F = 3(-y,x,0) from (a,0,0) to (a,0,2πb) along a straight line. 2. Do the same along a circular helix between the two points, parameterised as r = (a cosλ, a sinλ, bλ) 3. Compute the curl of F. How does this relate to the two integral calculations...

1.Homework Statement (Parabolic Cylinder) find the area of the surface extending upward form x^2 + y^2 =1 to z = 1 - x^2 using line integral 2. Could some one please outline the method to solving this. I tryed using spherical corridinates but am unsure if this was correcect The...

Hiya, I've just done a line integral question, and the final part of the question is, "by applying a standard test, determine whether the value of the line integral depends on the path followed between the given initial and final points." The only standard test, I know, is dq/dx, and dp/dy to...

Homework Statement It can be shown that the line integral of F = xj around a closed curve in the xy - plane, oriented as in Green's Theorem, measures the area of the region enclosed by the curve. (You should verify this.) Use this result to calculate the area within the region of the...

Homework Statement If the vector field F(x,y,z) is everywhere parallel to R and C is a curve drawn on a sphere with the center at the origin, then: \int\stackrel{C}{} F . dR = 0 why? The Attempt at a Solution Im not exactly sure if I understand the problem, but this is the explanation...

Homework Statement What is the line integral of B between points i and f in the figure? Homework Equations Ampere's Law: ∫B∙dl = Ienclosed * μ0 note: the integral on the left is a line integral. The Attempt at a Solution I applied Ampere's Law. I know that the contribution...

Hello! I am working on atmospheric scattering and therefore I have to calculate the optical depth along a given path through the atmosphere. The integral I have to solve is this one: \int_{P_a}^{P_b} f ds \quad with \quad f:R\rightarrow R, h\rightarrow e^{(-h/H_0)} P_a und P_b are...

Homework Statement Compute the line integral of v = 6i + yz^2j + (3y + z)k along the path (0,0,0) -> (0,1,0) -> (0,0,2) -> (0,0,0). Check your answer using Stokes' Thm Homework Equations The Attempt at a Solution I've tried breaking into three pieces. The first with dx = dz =...

i would like to find the area bounded by the curve (((x^2)/(a^2))+((y^2)/(b^2)))=xy/(c^2) i used the substitution given x=(ar)cos(theta) and y=(ar)sin(theta) i get : (r^2cos^2(theta)+r^2sin^2(theta))^2=xy/(c^2) thus r^4=xy/(c^2) substituting x=(ar)cos(theta) and...

Homework Statement Let F=x^{2}i+2xyj, and let C be the lower half of the unit circle, with perametrization r(t)=<cos(t),sin(t)>,\pi\leqt\leq\pi. Evaluate \ointF\cdotdr. Homework Equations The Attempt at a Solution The first thing I tried to do was to find a function f(x,y) so...

[SOLVED] Simple parametrized line integral This is from an example in my textbook. They want you to evaluate the line integral: \int_{C} y dx + 2x dy for the straight line segment in the plane from A(1, 1) to B(2, 4). The example says that this segment can be parametrized as x = 1 +...

Homework Statement Part 1: A 160lb man carries a 25lb paint can up a spiral staircase, which has radius 20 feet, completes 3 revolutions, and has final height 90 feet. What is the work done? Part 2: This time, the man's paint can leaks at a constant rate such that he loses 9lbs of paint...

Homework Statement Let x = \cos^{3} t and y = \sin^{3}t ( 0 \leq t \leq 2 \pi ). Also \rho(x,y) = k . Find I_0 = \int_{C} (x^{2} + y^{2}) \ dm Homework Equations The Attempt at a Solution So m = \int_{C} k \ ds = 3k \int_{0}^{2 \pi} \cos t \sin t \ dt . Then dm = 3k \cos t \sin t \...

Compute the line integral \int_{C} F\cdot dr where F = -y i + x j. The directed path C in the xy-plane consists of two parts: i) a left semicircle from (0, -1) to (0, 1) with center at the origin, and ii) a straight line segment from (0,1) to (2,1). i) r(t) = cos t i + sin t j [pi/2 <=t<=...

Suppose that F is an inverse square force field; this is, F(r) = cr/ |r|^{3} for some constant c, where r = xi + yj + zmbfk. 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 these points to the origin. Not exactly...

Dear Users Please help me in starting this problem I have tried my best but all in vain Calculate line integral v=X^2{x(Cap)}+2yz{y(Cap)}+y^2{z(Cap)} from origion to point (1,1,1) by three different routes (a) (0,0,0)→(1,0,0)→(1,1,0)→(1,1,1) Now there are three parts in this problem.I want...

Homework Statement This comes from Mathematical Methods for the Physicist from Susan Lea, Chapter 1 Question 25 Part B (Incase anyone is familar with the book). The question asks to evaluate the line integral integral (u*dl) where the vector u is: u = x*y^2 i + y*x^2 j along the...

Homework Statement Let C be the ellipse with center (0,0), major axis of length 2a, and minor axis of length 2b. Evaluate \oint_C xdy - ydx.Homework Equations I solved this two ways. First I parameterized x and y as x=a \cos \theta and similarly for y. I also applied Green's theorem, which...

I am confused about how \int_C f(x,y) dx = \lim_{||P|| \to 0} \sum_{i = 1}^n f(x_i^*,y_j^*) \Delta x_i is different from \int f(x,y) dx where P is a partition and its norm is the length of its largest elements. The index i represents an element in that partition and the asterik means...

Homework Statement F = <x^2 y^3 z, sin(xyz), xyz> S is part of the cone y^2 = x^2 + z^2 that lies between y = 0 and y = 3. Oriented in the direction of the positive y-axis. Homework Equations The Attempt at a Solution I know how to do the integral, and I get the correct answer except it's the...

Homework Statement Solve I = \int_{\gamma} f(z) dz where \gamma(t) = e^{i \cdot t} and 0 \leq t \leq \pi Homework Equations Do I use integration by substitution?? The Attempt at a Solution If I treat this as a line-integral I get: I = \int_{a}^{b} f(\gamma(t)) \cdot \gamma'(t)...

Homework Statement A curve is formed by the intersection of y^2/9 + z^2/4 = 1 and the plane x-2y-3z = 0. The particle moving along the curve goes from (6,0,2) to (-6,0,-2). Find the work done on it by the vector field F(x,y,z) = -yi + xj + yzk. Homework Equations I'm going to need to...

Int ((2xe^y)dx + (x^2e^y) dy) from (0,0) to (1,-1) I get the answer 2/e, while my book says 1/e. Am I right or wrong?

Hello everyone~ I have the following problem, its done in the book but I'm lost on how they came to the final answer. Evaluate the line integral: Integral over C xy dx + (x-y) dy, C is the line segment from (0,0) to (2,0) and (2,0) to (3,2). C = C1 + C2; On C1: x = x, y = 0; dy...

Hello everyone I'm confused on this line integral. The substiution is easy but I'm not sure where 2t is coming from... integral over C x^2*y*sqrt(z) dz; C: x = t^3; y = t; z = t^2; 0 <= t <= 1 integral over C x^2*y*sqrt(z) dz = integral 0 to 1 (t^3)^2 (t) sqrt(t) * 2t dt =...

Question 1) Integrate y^2 dx + zx dy + dz, along the circle x^2 + y^2 – 2y = 0 to (1, 1, 0) I am not sure how to begin on this problem. Would it be beneficial to convert to polar coordinates? The thing that is throwing me off is the fact that the circle’s equation is not the normal circle...

Homework Statement This is my problem: Compute the following three line integrals directly around the boundary C of the part R of the interior ellipse (x^2/a^2)+(y^2/b^2)=1 where a>0 and b>0 that lies in the first quadrant: (a) integral(xdy-ydx) (b) integral((x^2)dy) (c) integral((y^2)dx)...

Homework Statement Find the work done by the force field F(x,y) = x*sin(y) i-hat + y j-hat on a particle that moves along the parabola y = x^2 from (-1,1) to (2,4). 2. The attempt at a solution Please see attached file. The answer I get seems really, really, complicated and I have a...

Given F = iy - jx (this is my first post; not sure how you do vector notation here but I'm showing vectors in bold - hope that works). The problem is to show that this is a non-conservative force by integrating from the origin to (1,1) (ie, the path is y=x), and then do it again from the origin...

In my emag course we are reviewing vector calculus. I've forgotton a lot over the summer, so I just want to make sure I'm doing this properly. question) \vec E = \hat x y + \hat y x Evaluate \int \vec E \cdot d\vec l from P_1(2,1,-1) to P_2(8,2,-1) along the parabola x = 2y^2 . sol)...

Hello, I want help in the line integration of: Integral( 1 dy + 3 dx ), over the curve C. Where C is the union of two line segments: Line 1 from point (0,0) to (1, -3) Line 2 from point (1, -3) to (2,0) The thing is I do not know what to do with the integrand being composed of...

Line Integral and Magnetism Help! Here is a problem given to me as a takehome quiz I am having trouble even starting being that I missed that day we went over line integrals. Any hints or tips are much appreciated. http://ez-files.net/download.php?file=4ec07c5df1e8223123523fb11f038d39"

Suppose F=F(x,y,z) is a gradient field with F=\nabla f, S is a level surface of f, and C is a curve on S. What is the value of the line integral \int_C F dr? I know the answer is 0 but I cannot visualize why it would be zero?

http://www.mrnerdy.com/forum_img/ask.JPG is the first stepc orrect? TQ.

I have been working on the following line integral: \int_{T}^- {(-x^2y)dx + (y^2x)dy} where T is the closed curve consisting of the semi-circle x^2 + y^2 = a^2 (y>0) and the segment (-a,a) I will tackle this in two steps: 1) solve x^2 + y^2 = a^2 (y>0) for y and substitute...

Hi all, (This is part of a DJGriffiths, 3rd ed., problem: Prob. 1.28) Line Integral: Int(yzdy) [lower limit = (1,1,0); upper limit = (1,1,1)] but y does not change and is supposed to be integrated, while z changes and is not integrated. I have 2 questions: [1] I take z out of the integral...

Hi, I'm having trouble with the following question. Q. Let p be a real constant and \mathop F\limits^ \to = \left( {yz^p ,x^p z,xy^p } \right) be a vector field. For what value of p is the line integral \int\limits_{C_2 }^{} {\mathop F\limits^ \to \bullet d\mathop s\limits^ \to } =...

Hi, I've just started working on line integrals and I don't understand one of the examples in my book. \int\limits_C {y^2 dx + xdy} Where C is the arc of the parabola x = 4 - y^2 from (-5,-3) to (0,2). The book proceeds by suggesting that y is taken as the parameter so that the arc C...

I know this is dumb question but for some reason I have not been able to get the right answer to the following problem: \int_{c} 2xyzdx+x^2 zdy+x^2 ydz where C is a curve connecting (1, 1, 1) to (1, 2, 4). My parametrization is (1, 1+t, 1+3t). My limits are the problem...I think. By...

Calculate the line integral of the function v=(y的平方, 2x(y+1), 0) from the point a=(1,0,0) to the point b=(2,2,0) The correct Ans: 11 However, when I was calculuating the problem, I supposed(is that the right word?) to make the parameter equations, x=1+t y=1+t z=0 where 0<t<1...

I understand how to evaluate a line integral, but I don't know what it represents geometrically. Say you have \int_{C} x^4yd\mathbf{s}. What does this mean geometrically? I can see that \int d\mathbf{s} is the length of the arc (am I correct?), but I just can't seem to figure out what...

I'm having trouble on a line integral. Assuming that the closed curve C is taken in the counterclockwise sense. Use Green's Theorem. \int_C F\bullet dR where F=(x^2 + y^2)i + 3xy^2j and C is the circle x^2 + y^2 = 9 This is what I have done so far... \int_0^{2\Pi} \int_0^3 \-r^2...

Greetings again, Show that for F(x,y)=<2xy-3, x^(2)+4y^(3)+5> the line integral F(x,y).dr is independant of path. Then evaluate the line integral for any curve C with initial point (-1,2) and the terminal point (2,3). Thanks again, you all have been very helpful.

Greetings All Again, I wanted to thank you for the reply on my other problem, it was indeed very helpful and this is a very strange problem. So here goes : Compute the work done by the force field F(x,y,z) = <4y,2xz,3y> acting on an object as it moves along the helix defined...

I have a question which asked me to evalute the line integral around the curve x^2+y^2=r^2 (z=z0 (a constant)) of the following vectors: (0, z^2, 2yz) and (yz^2, yx^2, xyz) the first one I get as 0, and the second one I get as: -pi(r*z0)^2 Those answers I'm pretty sure are right...

Okay, I've searched PF. I actually found a thread that confirmed some of my assumptions. I've searched the web. But I still want to know what the geometric interpretation of a line integral with respect to x (or y) is. The example that made me want to know was \int y^2 dx + x dy ; It was...

My problem has force decreasing with F=(1/r^2)r, where F is a vecotr and r is unit vector. i need to find a).work done in moving from a point at r=sqrt(2) to a point at r=2*sqrt(2) by a direct radial path and (b) by a path from (1,1)-->(2,1)-->(2,2). Compare my answers. a)I did direct radial...

Hi All, I have the following electric field E = c(2bxy, x^2+ay^2) where a,b and c are constants 1. I need to find the line integral \oint E \cdot dl where the close integration path is defined by the triangle (0,0) (1,0) (1,1) 2. compute the surface integral \int\nabla \times E \cdot...

Heya! I was hoping someone could clear this up for me: how would a line integral be represted graphically? I've always liked calculus because it's easy to visualize (almost all the problems have graphs associated with them) - but I don't quite get how to visualize a line integral. Or is it...