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The line integrals I have been working with are expressed sometimes in this form: $$1)\int_c xy^4\,\,ds$$ where $c$ is a curve such as part of a circle and $ds$ is replaced by the arc length formula: $$\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}\,\,dt$$ and other times are expressed in this form: $$2)\int_c (x^2y+siny)\,dy$$ where $c$ again is some curve.

In both cases, if I'm correct, you parameterize the curve, substitute $x$ and $y$ for their parametric equations ($ds$ in $(1)$ becoming the arc length formula), and then you can evaluate the integral. What I don't understand is what the actual difference between these two ways of constructing the integral is. Can you write any problem in either of these ways? I understand that the first way is essentially taking small subarcs in two dimensions and then evaluating the function at each one (and summing), but I don't understand visually or intuitively what the second one represents. If anyone could give me more intuition I would really appreciate it.

fmtcs
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  • The second one is a vector line integral $\int_C \vec{F}\cdot d\vec{r}$ where $$\vec{F} = (0,x^2y+\sin y)$$ and $d\vec{r} = (dx,dy)$ – Ninad Munshi Jul 27 '21 at 17:42
  • please see if this helps: https://www.khanacademy.org/math/multivariable-calculus/integrating-multivariable-functions/line-integrals-for-scalar-functions-articles/a/line-integrals-in-a-scalar-field – Math Lover Jul 27 '21 at 17:50

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