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I came across the trapezium rule and started wondering: if definite integration can be defined as an attempt to approach an infinitely fine approximation of the area under the curve, can the area under the curve be approached through different geometric ways than an infinite number of rectangular or trapezoidal stripes? I tried to investigate the question on my own by dividing the area under the arbitrary function, f(x), in the interval [a,b] into an infinite number of circles that are tangent to one another and as the limit approaches infinity there would be an infinite number of circles. However, I struggled to solve the first lagrangian multiplier:

Max $R^2$=$(x-x_0)^2+(y-y_0)^2$ s.t. $a≤x≤b$, $0≤y≤|f(x)|$

The idea is that if the lagrangian is solved, it would act as a constraint on the second largest circle in the region, and so on, and so forth.

Is my method correct? I know this is cumbersome and quite inefficient, but can integration be defined in terms of this way? Are there certain advantages to this approach given the geometric properties of the area in question? Thanks.

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    $\begingroup$ Usually one proves that if a function is Riemann integrable, it does not matter what subdivisions of the interval you choose as long as the widths of all the subintervals go to zero. $\endgroup$
    – Ross Millikan
    Feb 6 at 3:19
  • $\begingroup$ Thanks for the comment. What exactly do you mean by Riemann Integrable? $\endgroup$
    – Daniel
    Feb 6 at 5:03
  • $\begingroup$ There are a number of definitions of definite integrals. The first one usually taught is the Riemann integral A function that can be integrated that way is said to be Riemann integrable. Another is the Lebesgue integral which can integrate a somewhat different set of functions. The integrals agree where both can do them. $\endgroup$
    – Ross Millikan
    Feb 6 at 5:41

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