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Questions tagged [line-integrals]

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used.

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral as well, although that is typically reserved for line integrals in the complex plane.

The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve).

Read more on wikipedia's entry Line integral.

1126 questions
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Finding path for maximum value of line integral

I need help in the following question: I have a field: $$F=\left(y^3-3y+xy^2,3x-x^3+x^2y\right)$$ bounded in region $D$ defined by $x^2+y^2\leq 2.$ I need to find a path $C$ that goes from $(1,1)$ to $(-1,-1)$ inside $D$ such that the value of the…
Frogfire
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How do I parametrize this vector so that I can evaluate this line integral?

So I'm asked to evaluate the line integral $$ \int_C F dr $$ where $$ F = yz\cdot\vec{i} + \sin(y)\cdot\vec{j} + \cos(z)\cdot \vec{k} $$ along the curve $$r(t)=t^2\cdot\vec{i}+t\cdot\vec{j}+t^3\cdot\vec{k} \quad \text{ for } 1 \leq t \leq 3 $$ My…
tsp216
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Line integral, not depend on orientation?

Let $A=(0,0)$ and $B=(0,1)$. Using $r_1:[0,1]\longrightarrow\mathbb{R}^2$. $r_1(t)=(0,1-t)$ $$\displaystyle\int_B^A 1=\int_0^11\,dt=1.$$ On the other hand, Using $r_2:[0,1]\longrightarrow\mathbb{R}^2$. $r_2(t)=(0,t)$ $$\displaystyle\int_A^B…
yemino
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Line integral. Conservative vector field

Please help me with solution of the given problem: Let $\gamma_1$ and $\gamma_2$ be the circles of $R^2$ of radius $r=2$, centered at the points $(0,0)$ and $(8,6)$ respectively. Assume that $\gamma_1$ is clockwise oriented and that $\gamma_2$ is…
user351810
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Checking of a line integral calculation

This is from a past exam paper that I'm using for revision I've calculated the answer as $21.75$ but I'm not sure I've done it right. The question is a follows: Calculate the line integral $\int F.dr$ where $ F = (x^3,yx)$ and $y=3x$ also…
James Page
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Why is using the right endpoint of a rectangle advantageous when determining the constant in the anti-derivative of area function?

I am on the introduction of integral and antiderivative of finding area under curve. The book didn't tell me why (that's for later), but it only says that the anti-derivative the continuous function is the area under the curve. And when we take the…
most venerable sir
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Line Integral of given curves

The given vector field is $X(x,y,z) = (y,-x,1)$ on $\mathbb{R}^3$. Calculate $\int_{x_1,C}^{x_2} (X|dx)$ for the following curves. a. $\mu_1(t) = (\cos(t), -\sin(t), \frac{t}{2 \pi}), t \in [ 0, 2\pi]$ b. $\mu_2(t) = (\cos(t), \sin(t), \frac…
rndflas
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Line integral and attraction of a material point by a material curve in R^3

According to Newton's law of universal gravitation, a material point $P$ with mass $m$ attracts a material point $P_0$ with mass $m_0$ with a force directed from $P_0$ towards $P$, of size $k\cfrac{mm_0}{r^2}$ (further for simplicity k=1). Note that…
Paull
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Find the attraction of an infinite homogeneous line of a unit mass located at a distance h from the line.

We know that \begin{equation} X=m_0\int_{\sigma} \frac{f(P)\cos{\alpha}}{r^2} ds, \hspace{1cm} Y=m_0\int_{\sigma} \frac{f(P)\sin{\alpha}}{r^2} ds; \end{equation} where $r$ is the length of the vector $r=\overline{P_0P}$, and $\alpha$ - the…
Paull
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about line integral evaluation

Any hints on which equation to use is most appreciated! I give up on thinking at 5am using my brain which is made up of mashed potatoes :) $\int\limits_C \underline{G} \cdot d\underline{r}$ with $G(x,y) = xy \underline{i} + 2 y^2 \underline{j}…
username_1326
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all values of line integral

Find all values of the line integral $\int_{(1,0)}^{(2, 2)}{\frac{-y}{x^2 + y^2}dx + \frac{x}{x^2 + y^2}dy}$, by a path that does not contains the point $(0,0)$
Diego Campos Laboissiere Ville
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Tricky Polar Line integral problem

I need to calculate the following line integral over the curve C: $$\int_C e^{\sqrt{x^2+y^2}} ds $$ where $C$ is the circuit bounded by $r = a, \varphi = 0$ and $\varphi = \pi/4$. I tried separating the integral into these three parts and got an…
Nacho
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What's the reasoning behind the parameterization in this line integral?

I could not quite get why we have parameterized r(t) as in the picture below. In fact, parameterizing curves whenever I try to solve a problem remains to be the only step I'm having trouble with in line integrals even after reading many things on…
Ali
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Line Integral calculating the ds

Doesn't change the result, but why in the last steps constant in front of the integral is 9 and not 27? Where did the 9 that came from squaring the dx/dt and dy/dt go?
Ali
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Green's formula for area

Using Green theorem I need to calculate the area bounded with $(x+y)^4=x^2y$. First(after converting to polar coordinates) i get $x=\cos^6\phi\sin^2\phi$ and $y=\cos^4\phi\sin^4\phi$. And after i plug that into Green's formula i get…
Trevor
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