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On the topic of integrals, why would integrating this figure from $$\int_{a}^{b}\textbf{F}(x,y)ds$$ give the area under the curve and not the area over the curve? Would simply integrating $\int_{b}^{a}$ give the area over the curve?

enter image description here

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    $\begingroup$ Can you see that integration gives the area under the curve in two dimensions? $\endgroup$
    – Toby Mak
    Mar 18, 2020 at 23:38
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    $\begingroup$ As you wrote it, it's a line integral so it would be the area over the curve with $x$ fixed at $0$. A visual is a curtain over the curve with the height being the value of $F(0,y)$. The shaded region is $\int_0^2 F(0,y) dy$. $\endgroup$
    – ProfOak
    Mar 19, 2020 at 0:01

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