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Why integration is a inverse of differentiation ?? Explain geometrically. I know differentiation is a tangent line and integration is a parts of curves but why a tangent line and parts of curves are inverse of each other??

If $f(x) > 0$ is continuous, let $A(X)$ be the area under the curve $y=f(x)$ for $a < x < X$. Then
$ A(x+epsilon) - A(x)$ is approximately the area of a rectangle with base $epsilon$ and height $f(x)$, thus $$ dfrac{A(x+epsilon)-A(x)}{epsilon} approx f(x) $$
Now take the limit as $epsilon to 0$.

Here's an attempt at conceptualising (not proving) it geometrically.

Roughly speaking: suppose you start with a graph of $y=2x$ and let $A$ be the area under the section from $0$ to $x$. By integration you find that $A=x^2$.

Now you can draw a second graph, this time showing how the area varies with $x$. It will be a graph of $A=x^2$. If you want, you can differentiate this and find that the slope at $x$ is $2x$.

Now you've got two ways to see how fast the area changes with increasing $x$:

• look at the slope of the second graph. Or, in fact


• look at the height of the first graph.

How does this work? The area is built up of vertical slices, so how fast it grows is given by how tall they are.

Integration asks: "What happens if we add up the slices?" and differentiation asks "What size do the slices need to be?"