Intereting Posts

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i read about some interpretation ideas in Interpreting Line Integrals with Respect to $x$ or $y$

and i was wondering if the interpretation given below is right or not ?

informally put can we say $\int_C f(x,y)\,dx$ is the area of the shadow caused by the fence on the curve $f(x(t),y(t)$.

the light source is perpendicular to x-z plane and the shadow is also made in this plane.

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lets say we have : \begin{align}

\int_C f(x,y)\,dx &:= \int_\alpha^\beta f(x(t),y(t))\,\color{red}{x'(t)}\,dt,\tag{2}

\end{align}

we define a bijection ( one-to-one and onto function ) from set of all x’s belonging to the curve to the curve itself call it $g$. $g(x) = (x, y(x^{-1}(x))$. assuming we have functions $x(t), y(t)$ describing our curve and $x(t)$ is one to one and $x^{-1}(t)$ is the inverse function of x(t) ; then we have :

then :

so can we say the area in yellow (the shadow) is the interpretation of the \begin{align}

\int_C f(x,y)\,dx &:= \int_\alpha^\beta f(x(t),y(t))\,\color{red}{x'(t)}\,dt,\tag{2}

\end{align}

just like the area in blue is the interpretation of the \begin{equation}\tag{1}

\int_C f(x,y)\,ds := \int_\alpha^\beta f(x(t),y(t))\,\color{blue}{\sqrt{[x'(t)]^2+[y'(t)]^2}}\,dt

\end{equation}

also can we say \begin{align}

\int_a^b f(g(x))\,dx &:= \int_\alpha^\beta f(x(t),y(t))\,\color{red}{x'(t)}\,dt,\tag{2}

\end{align}

Ps: we can find function g if the curve C is one to one with respect to x axes .

but i suspect if that’s no the case just like a normal shadow the overlapping parts would not be calculated again an the resualt would be the area of the shadow!

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