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## Main Question or Discussion Point

For showing how the integral is the area under the curve, i know how to drive the relation [tex] \lim_{x_k\rightarrow 0} \sum_{k=1}^\infty f(x_k)dx_k[/tex]. But i don't know how this is the integral itself. I know that i can try specific examples, use the sum, take a finite interval and show that it is approximately similar to its integral. But is there anyway to prove it generally? Thanks alot.