I am learning about finding the area under a curve and was wandering what $dy$ had to with it. When finding the area under a curve you want the anti-derivative, so you get $\int f(x)dx$ but if $F'(x) = f(x)$ then would $f(x) = \frac{dy}{dx}$? If this is correct (if not please explain where I made a mistake(s)) then $f(x)dx = dy$. How does $\int dy$ result in the area under the curve of $f(x)$?
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Under which curve: $y=f(x)$ or $y=F(x)$? – zipirovich Sep 06 '18 at 22:38
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under the curve of f(x) – Kile Maze Sep 07 '18 at 00:33
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Then, according to what you said yourself, $f(x)=y$, not "$F(x)=y$". And therefore, to reiterate: $f(x)=y$, not "$f(x)=\frac{dy}{dx}$". – zipirovich Sep 07 '18 at 03:10
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Too long for a comment.
I think you are trying to understand the fundamental theorem of calculus, which says, in a way, that there are two equivalent ways to find the area under a curve. One si by calculating a definite integral as a limit of sums, the other is by finding an antiderivative.
Here are two links that might help.
Why can't the second fundamental theorem of calculus be proved in just two lines?
(Marking as community wiki since it's not really an answer, but may point to one.)
Ethan Bolker
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