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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.

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Defining a curve to solve a line integral

$$\int_C z e^ {-z} dz$$ with $C$ being any curve from $i$ to $1 + i$ I defined $C$ as $$z(t) = t + 1 $$ $$i \leq t \leq 1 + i $$ Such that $$\int_C f(z)dz = \int_{i}^{1+i} (t + 1)e^{-(t+1)}dt $$ $$=e^{-2-i}-e^{-1-i}$$ Does that seem correct? Is…
paranoidhominid
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How to evaluate line integrals without using Green's Theorem?

I recently learnt about line integrals and Green's Theorem. But the lecturer gave us an assignment to answer how to calculate line integrals "directly without using Green's Theorem". I've looked at the notes, but I can't seem to see the difference…
Hani Zahrul
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Line Integral with Large Radicals

The integral of $x^{1/33}+y^{1/27}+z^{1/39}$ of the line segment $(161, 283, 73)$ to $(168, 361, 145)$. I tried to do it on my own but my answer $-2873.78$ seems extremely wrong. Originally I tried to brute-force it. I did…
James
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Mass of wire using line integral

If $\rho(x, y)$ is the density of a wire (mass per unit length), then $m = \int \rho(x,y)ds$ is the mass of the wire. Find the mass of a wire having the shape of a semicircle $x = 1 + \cos(t), y = \sin(t)$, where $t$ is on the closed interval from…
user3032755
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Evaluate line integral using green's theorem

I'm having a little problem understanding this question on Green's Theorem. It asks to use Green’s Theorem to evaluate the line integral $\int_C F.dr$, where $F=(e^{y^2} −2y)i+(2xye^{y^2} +\sin(y^2))j$ and C goes along a straight line from $(0,0)$…
thbcm
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Line integral of scalar field

I couldn't develop the parametric curve nor I didn't understand very well This is the question: There is the answer: I tried $\int_{0}^{1}(3y-\sqrt{z})ds = \int_{0}^{1}(2y)ds$ because $z=y^2$ but didn't work I used first this parametric: A(1,0,0)…
Goldman7911
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Show that a line integral is path independent?

I have to show that a line integral is path independent between two points, and while I know how to check if one is, I have no idea where to begin proving that one is. The equation looks very simple to integrate, but I don't even know where to…
tokola
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Calculating line integrals using Stokes theorem

Use Stokes' Theorem to find the exact value of the line integral $$\int_{C}(y\:dx+z^2\:dy+x\:dz)$$ Where $C$ is the curve of intersection of the plane $2x + z = 0$ and the ellipsoid $x^2 + 5y^2 + z^2 = 1$, oriented counterclockwise as seen from…
ys wong
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Confusion on double integral

If i do closed integral($dxdy$) in Cartesian coordinates and try to find area of a circle by putting $y=\sqrt{(a^2-x^2)}$ then $dy$ becomes $y$ and then its closed integral($ydx$) and the double integral becomes $0$ for closed integral in cartesian…
Souvik
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Why this line integral over this straight line isn't possible without parametrization.

There is a line integral over a straight line as follows (the problem here: https://tutorial.math.lamar.edu/Solutions/CalcIII/LineIntegralsPtI/Prob1.aspx): Line Integral: $$\int 3x^2 - 2y\ ds$$ Equation of the line: $$2y=7x-9$$ In the solution, The…
Hooman
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What did James Stewart mean by "the line integral reduces to an ordinary single integral in this case"?

What did James Stewart mean by "the line integral reduces to an ordinary single integral in this case"? See the para. aside my two green question marks below. How do you symbolize "the line integral reduces to an ordinary single integral in this…
user1124753
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Evaluating Line Integral for Several Cases?

Given $F=(3x^2+6y)\mathbf{\hat{x}}-14yz\mathbf{\hat{y}}+20xz^2\mathbf{\hat{z}}$, I am trying to evaluate the line integral $\int_C{A}\cdot d\mathbf{r}$ from $(0,0,0)$ to $(1,1,1)$. C is given as three different cases, where $x=t, y=t^2,…
A4Treok
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