No use of the tangent in Riemann sum?

[Juwane]

Where is the use of the "tangents at every point on the curve" in the Riemann sum? Riemann sum allows us to find the area of under the curve, and this involves only the height of each rectangle (i.e. the function f(x) at each x), and the width (i.e. the x), and the two are multiplied together. This is done for every point x, and in the end all the products are added to give the final area. That's it--that's how the area under the curve is found. Where did we use the tangent in this, that they should become so important in calculus?

 

[JSuarez]

Tangents, or the regularity conditions for their existence, are not necessary for integration (remember that a function don't need to be differentiable to be integrable, but it does to have a well-defined tangent).
The Theory of Integration may be entirely developed without mentioning derivatives (i.e. tangents), and Differentiaton may, conversely, be developed without mentioning integration.
The fact that the two are related, by the Fundamental Theorem of Calculus, is a deep fact.

 

[Juwane]

So no tangents in integral calculus. But is differential calculus wholly based on tangents and does not have to do anything with finding areas?

 

[JSuarez]

The full answer is, of course, much more complicated, because the history behind it is complex, but the main problem of Differential Calculus is the determination of rates (that is, derivatives or tangents), where in Integration, the problem is the determination of content (lenghts, areas, volumes, etc.).
Nowadays, it the Fundamental theorem, it's usual to treat them as one subject.

 

[Anonymous217]

Well, for one example of its use, tangents are used for to find the length of a curve.

 

[HallsofIvy]

Well, for one example of its use, tangents are used for to find the length of a curve.

?How? Every derivation of the formula for arclength I have seen uses secants, not tangents.