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.

603 questions
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A given perimeter length that is circular encloses the maximum area - which are the (analytic) proofs? [duplicate]

I'm guessing Newton, because of his integrals. But what proofs have been established, and which is the most mathematically intuitive one? I was looking for the tag "circumference", supplied the newer ...
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How to compute the line integral of $\int_{C^+}(x^2-y)dx+(y^2+x)dy$?

I need help with this problem: Compute the line integral$\int_{C^+}(x^2-y)dx+(y^2+x)dy$ where $C^+$ is the parabollic arc $y=x^2+1$, $0\leq x\leq 1$ oriented from $(0,1)$ to $(1,2)$. First I ...
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Finding the maximum value of $\oint_C \vec{F} \cdot d\vec{r}$ of a vector field

I'm stuck with trying to find the maximum value for $\oint_C \vec{F} \cdot d\vec{r}$ of the vector field $\vec{F}=\langle5z, x, y \rangle$ where $C$ is a simple closed curve in the plane $2x+3y+z=7$. ...
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Integrating Factors Via Minimization

I am trying to algorithmize an idea to find integrating factors for inexact differential equations via optimization. Currently, I'm sticking to simple cases that can already be solved by other means. ...
Why the weird $\partial{Q}$ notation for the integral region for Green's Theorem? $$\int_{\partial{Q}} W \cdot ds = \iint_Q \frac{\partial{g}}{\partial{x}} - \frac{\partial{f}}{\partial{y}} dx\ dy$$ ...