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Is there an first-principles equation for integration that can be explicitly solved like that for differentiation? - I'm trying to understand Integration intuitively. Thanks heaps.

I've tried to piece together one but can't quite solve it (for simple functions like f = x). Things cancel and disappear and I get obviously wrong results.

[itex]\int^{x_{2}}_{x_{1}}F(x)dx[/itex]=lim[itex]_{n\rightarrow\infty}[/itex][itex]\sum^{n}_{i=0}[/itex]F(x[itex]_{1}[/itex] + i[itex]\Delta[/itex]x)[itex]\Delta[/itex]x in which [itex]\Delta[/itex]x=[itex]\frac{x_{2}-x_{1}}{n}[/itex]

I want to know how integration was derived and how it works, like I do with differentiation and limits.

Thanks for reading my post.

I've tried to piece together one but can't quite solve it (for simple functions like f = x). Things cancel and disappear and I get obviously wrong results.

[itex]\int^{x_{2}}_{x_{1}}F(x)dx[/itex]=lim[itex]_{n\rightarrow\infty}[/itex][itex]\sum^{n}_{i=0}[/itex]F(x[itex]_{1}[/itex] + i[itex]\Delta[/itex]x)[itex]\Delta[/itex]x in which [itex]\Delta[/itex]x=[itex]\frac{x_{2}-x_{1}}{n}[/itex]

I want to know how integration was derived and how it works, like I do with differentiation and limits.

Thanks for reading my post.

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