I read in Calculus made easy book(Thompson) that Integration is adding up things( Summing of Infinite Infinitesimals. But is this correct? Because very few books give this definition whereas most books introduce integration as finding the area under Curve which is infact one of the applications of integration. Moreover I read on net that historically Integration was invented first and later the concept of derivative was developed.

Then how were Integral formulae found before Derivative concept was developed?

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    $\begingroup$ You don't need the derivative to find the area under a curve. Indeed, the definition I've learned in school is: Determine the area under the curve by cutting it into slices, approximate the slice by a rectangle, add them up, and take the limit of slice width to zero. Which actually combines both mentioned definitions: Area under the curve, and adding things (namely, areas) up. Note that there's no derivative (slope of the tangent) involved. $\endgroup$ – celtschk Sep 13 '12 at 15:48
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    $\begingroup$ Limits of what would later be called Riemann sums, and relatives. It all goes back to Archimedes and even slightly before. Substantial achievements in Kerala, $14$th century and later. The indivisibles of Cavalieri. Fermat's Method for $x^r$ where $r$ is positive rational. Many other $17$th century "quadratures" before calculus was officially born. $\endgroup$ – André Nicolas Sep 13 '12 at 15:51
  • $\begingroup$ I'm Sorry I should have been much more clearer w.r.t my question. What I mean is integral cosx is sinx bcoz derivative of sinx is cosx. How was integral cosx (not only cosx, for that matter any other formulae) found before the advent of differential calculus? $\endgroup$ – Ranjan Yajurvedi Sep 13 '12 at 16:03

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