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.

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Using a line integral to finding a function for the distance between two points on a sphere

Per the title, suppose in R 3 we have a unit sphere. Two vectors intersect that sphere at two points. I know the distance between two points can be found as a function of the angle between the two ...
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Finding parameterization for exponential integrand

I'm currently working on a practice problem for my Calculus III class: "Evaluate the line integral along the negatively-oriented closed curve C, where C is the boundary of the triangle with ...
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Visual Representation of the Line Integral

I am trying to better understand the visual interpretation of a Line Integral. We are all told that a classic integral of a function represents the area under that function between two certain points -...
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Using Line Integral to Find Work done by a Force along a Curve

[Find the work done by the force $\boldsymbol{F}=-10y\boldsymbol{i}+4x\boldsymbol{j}$ along one loop of the curve $r=\sin(9\theta)$] I have some trouble formatting/understanding the question. I have ...
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How to transform area integral $\int_{D} \omega^2 \ dx \ dy$ into boundary integral $\oint_{C} \square \ ds$?

Let $\omega$ be a function that satisfies the Laplace's equation $$\nabla^2 \omega = 0$$ The values $\omega$ and $\dfrac{\partial \omega}{\partial n}$ are known in the boundary, but not in the ...
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Proof of Exactness of Closed Forms Using Leibniz' Rule Given Parametrized Curve in R^2

I have run into trouble with a foundational proof regarding the exactness of closed forms which are $C^1$-differentiable. I have seen other proofs utilizing integration factors and I see why the ...
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Line Integral along a Ellipse over a Scalar field

Given is the Line Integral: $$\int_C = \sqrt{\frac{a^2y^2}{b^2} + \frac{b^2x^2}{a^2}}ds$$ the Path $C$ is along the border of the ellipse with: $$\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$$ and moves ...
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Vector Line Integral For Biot Savart Law

How would one go about computing the vector line integral presented in the Biot-Savart law: $$\vec{B}=\int_c\frac{\mu_0I}{4\pi} \frac{d\vec{l}\times\hat{r}}{r^2}$$ I know how to compute vector line ...
We know that $$X=m_0\int_{\sigma} \frac{f(P)\cos{\alpha}}{r^2} ds, \hspace{1cm} Y=m_0\int_{\sigma} \frac{f(P)\sin{\alpha}}{r^2} ds;$$ where $r$ is the length of the ...