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I have know that both integration and differentiation are related and how they are both inverse functions of each other but how does integration which is area under the curve really related to differentiation which is slope of the curve what is the relation between them geometrically and if possible a proof and a theorem for it on how they both are related geometrically

Vignesh
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  • I think this answer is helpful https://math.stackexchange.com/questions/15294/why-is-the-area-under-a-curve-the-integral/15302#15302 – Ajin Shaji Jose Jan 17 '24 at 13:01

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Suppose you are integrating a continuous function $f$ from $0$ to $x$. Now add an infinitesimal increment $dx$ so that you are now looking for the area under the graph of $f$ between $0$ and $x+dx$. The difference between the old area and the new area is the infinitely thin vertical "rectangle" whose height is appreciably constant - namely, it deviates from $f(x)$ only by an infinitesimal amount. Thus the area of the rectangle is the product $f(x)dx$. If you want to differentiate the integrated function at the point $x$, you will be dividing $dy$ which is the area of the thin rectangle, by the $dx$ which is its width. Therefore what you get back the height, i.e., the value of the original function at $x$, namely $f(x)$.

Mikhail Katz
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