Скачать или смотреть Riemann sums and integration are bothmethods úsed to calculate the area under acurve,

Riemann sums and integration are both

methods úsed to calculate the area under acurve, but they represent different stages ofthe same idea. A Riemann sum is anapproximation: you divide the interval intosmaller subintervals, draw rectangles (or0other simple shapes) that touch the curve ina chosen way (left endpoints, rightendpoints, or midpoints), and then add upthe areas of these rectangles. The accuracyof this method depends on how manyrectangles you use and how they are placed.With only a few subintervals, the estimatecan be rough, but as the number ofsubintervals grows, the approximation gets
subintervals grows, the approximation getscloser to the true area.

Integration, on the other hand, is the precisemathematical limit of these Riemann sums asthe width of the subintervals approacheszero. Instead of approximating with finitelymany rectangles, integrationcaptures theexact area by considering infinitely manyinfinitesimally thin partitions. This transitionfrom approximation to exactness is one of 265the central ideas of calculus. While Riemannsums are useful for visualizing andunderstanding the process, integrationprovides a powerful and exact tool for
understanding the process, integrationprovides a powerful and exact tool forsolving problems across physics,engineering, and mathematics.


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